Updated: Tuesday 16 August 2022.

Bike commuting in rain is a lengthy topic. Arguably, an instructional video would be better, but I cannot record myself over an entire rain-season, just to accumulate shots. And people might not have the patience to sit through it.
So, putting their (theoretical) laziness together with my decidely-not-theoretical laziness, the following may become my prosaic outline, once I become arsed to do it!

Proviso

As in all things on this site, Canadian experiences dominate my perspective. No, not the Arctic, but it does cover a range of +43 C to -47 C.
As pertaining to _this_ article, the rain has varied from warm downpours to very cold, heavy rain.
I’ve never had to shower when arriving at work. My decidedly unscientific suspicion is that, for most people, the notion of ‘having to shower at work’ has to do with poor skills than with their physiology. Your physiology might demand otherwise, but I suspect that there are a lot of people who lack the skills to avoid doing that.

The Bike

You need fenders. If you’re going to ride in rain, your bike needs fenders. Period.
Other than that, a hybrid bike is an all-around solution for most people. The tyres should be neither slick/smooth (like a road racer’s) not massively knobby (like a mountain bike’s); but they must be knobby, for good traction in rain.

Venting

Bike commuting, and indeed any non-naked exercise in the outdoors is about layer management. (Or, one may call it temperature management, though that could be misconstrued as somehow managing the outside temperature! And, you can’t really manage your body temperature _directly_ anyway. ) It’s about how you break down your clothing into a set of separable items, each of which has a different contribution to your comfort, protection and temperature management, and which can be tweaked as needed.
The tweaking often involves venting. This refers to preventive venting of heat and perspiration. Explicit examples are zips and breathable fabrics.

Breathable Fabrics

Fabrics which allow perspiration and heat to exit your clothing constitute passive venting.
No, I am not talking about wearing only synthetics or merino wool; those are a given, but not the topic at hand. Pretty much any fabric is likely to be breathable, except for shells such as nylon, rubber et al. Plain rain-shells do not breathe. Wear a traditional wind-breaker, or rubber boots/gloves, and you’ll soon be accumulating sweat after a bit of exercise!
The point here, in this context, is that your rain shell should be a WPB (WaterProof Breathable)–namely, Gore-Tex and its numerous clones. This applies to hats, gloves, pants, jackets and even footwear.
No, such shells are not as breathable as a plain, synthetic or wool fabric, but they are _far_ better than old-fashioned rain-gear. No-one skis or snowboards with those things anymore, except ironically–to wit, Eighties ski gear! So you should not bike in cold rain without WPBs, either.

Zips

More common in shells, pit zips and front zips are prime examples of active venting. The harder the shell, (the less breathable it is, and so) the more crucial these are. The less experienced the rider, (the more likely he/she/per will neglect to monitor per body temperature, and so) the more helpful these are.
The most important is a pit zip. This is in the arm pits, allowing a large outflow of heat and perspiration, and sometimes a large inflow of cold air, directly from/to your torso. Fantastic for creating large deltas in your core temperature!
Front zips (not buttons) are equally essential. Even in mid-ride (given a safe context), I have used them to manage my temperature. If not dumping a large amount of heat by unzipping all the way, at least by pre-empting the need via partial-unzipping well in advance.
Other zips may be present, as in the thigh zips common on snowboarding pants.

Two-way zips are the best, allowing more control over where and how and how much you’d unzip. (There are quite a few combinations of ‘two-way’ zippers. Unless if I can be bothered to make my own, I just pay close attention to what the clothing item I’m purchasing has.)

To illustrate their importance: Except in base layers, I never buy a pull-over! Even in non-shell fabrics, I aim for something tweakable. Unless if it’s purely a base layer, I look for a zipper or some such. Even in base layers, I prefer to at least have a zip at the neck, because if the context calls for it, I might use that to micro-tweak my temperature.
For thermals, too. Whether as a mid-layer or, given the context, as an outer layer, full, frontal zips are the very minimum I look for. Simply, it _has to be_ possible for me to dump large amounts of heat, if the need arises. Not that I usually allow things to get to that point, but the context may necessitate a large pre-emption.

All of the above is particularly pertinent to rain. Dry cold is far more manageable than snow, and snow is far more manageable than cold rain. Simply, cold rain is the single most challenging weather-condition to a bike commuter. The more sustained, the worse! If it’s light rain, but your ride is long, or it’s heavy rain during a brief ride, cold rain is a challenge. Monitoring your body temperature becomes crucial.

Core Temperature

Layering is about temperature management. I pay attention to two things: My core temperature, and my skin temperature. The former is that deep within my torso.
Surface temperature is more responsive to change; core temperature is large, sluggish and slow to respond to change. As such, I am constantly monitoring my core temperature, anticipating its direction. If it gets windy, or an uphill is approaching, or there is a large amount of snow to plow through, or I had neglected to pay attention to my core, or the shell was too zipped-up for too long, I will unzip large areas in order to dump a large amount of heat _quickly_. It may well be that my core temperature is perfect; but, seeing the uphill approach, I’d want to dump, to lower my temperature even further.

If all’s well, but my temperature is slowly rising to a bit above the chill-comfort boundary I like it to hover at, then I might unzip something just a little bit. Not to dump a large amount of heat, or to enjoy a large inflow of cold air, but just to micro-tweak my temperature, lowering it by a degree or two.
Or I might lower a neck gaiter, or take off the hat, or take off the gloves. Sometimes, I raise the _bottom_ of the neck gaiter, to lower my neck’s temperature, which is closer to my torso; or, perhaps, my face was too cold, so I’d wanted that to retain its temperature. I’ve been in situations in which I’ve had to dump heat, but could not stop, and could only safely take off _one_ glove! Even that can help, if the context is cold enough.

Layering Down

When tweaking won’t suffice, layering down is called for–a large dump of heat!
Again, this ought to be anticipatory, rather than reactive. When approaching my destination, I often try to coast the last kilometer or two, with reduced layers. But that applies more to _dry_ conditions.

In cold rain, layering-down is not always possible. When returning home, layering down may be fine; when approaching work, it may not. Really, in cold rain, I try to manage my temperature so well that I won’t need to layer down. By the time the latter is reached, you’ve lost the battle.

Pants

These have to be split-style: The front is a WPB; the rear is plain, non-waterproof, highly breathable fabric.
In other words, the front protects against the wind and the rain; the rear vents heat and any perspiration.

Ankles

In heavy rain, protection is needed for one’s ankles. Rain will pour down, over the front of your pants, and into your socks and ankles, then into your footwear. As a consequence, even if your footwear may be WPB, your feet will get soaked!
Unfortunately, unlike snowboarding pants, manufacturers are not producing any sort of integrated gaiters for rain pants. Regular gaiters, used for snow travel, are either oversized or undersized or potentially interfering with the drive train. I’ve made my own, to varying degrees of success.
Other than that, cyclists use booties–ugly, uncomfortable, cumbersome, often ill-fitting booties! You may think differently about them!

Jacket

For a casual ride in light rain, a simple wind-breaker might suffice; if warm rain, perhaps not even that! (With non-WPB wind-breakers, I’d still insist on at least a full, front zip.)
For cold rain, sustained or not, a WPB jacket is called for, with a full, front zip, and pit zips.
Chest pockets are a great idea, and side pockets are mandatory (to put-away gloves in).

Unlike with pants, a split-style jacket is not useful. Most of your time is spent leaning on the handlebar, so your back receives a lot of rainfall. (And there may be a backpack involved!)

Contrary to the practice prevailing among some, no long tail is needed. That is just _absurd_. Why would you need WPB to collect mud! A commuter bike needs fenders. Period!

Footwear

In warm rain, I wear sandals. The colder it gets, and the more exposure (in the duration of the ride, or in the amount of rain), the more likely that I will use WPB footwear.
Plain synthetic socks should do. But, I’ve long been in the habit of wearing wool socks.
I’ve not used WPB socks with plain (or WPB) footwear. The only times I use them are when I’m hiking in cold precipitation: I tend to hike in sandals; so, when there is lots of cold rain, WPB socks are a spare I carry with myself. Their seams are vulnerable, which is why I carry them as only an emergency spare.

Hat

When commuting, I do not wear a helmet. So, under cold rain, I use a WPB hat. The brim for this should be stiff, and neither thin nor wide: Too wide, or too soft, and the wind will blow it upwards; too thin, and it won’t cover my glasses. Actually, glasses are never really protected from rain; a stiff, baseball-cap-style brim might help, but I’ve rarely tried that.

Gloves

Layering mandates that integrated gloves be avoided. So, I don’t wear gloves which have an integrated layer of insulation. For cold rain, this is even more so.
What has worked for me is fleece gloves (though wool’d be better, albeit too delicate) under shell mittens. If my temperature rises, I take off the thermal layer (fleece).
Usually, the only product I promote on this site is my own: the M!nder bike alarm. But, credit is due to OR’s Revel Shell, and Taiga’s mitten shells. The latter I hoard, using it especially for skiing and snowboarding; the former might, admittedly, be more suitable to cycling, being softer and lighter. … Yes, the OR is for cycling; I’ve never used the Taiga for that, whereas the OR I always use so.

On WPB Shells

As explained elsewhere on this site, Water-Proof Breathable (WPB) shells comprise a laminate and a coating. The latter will wear off with time and washing; while they can, and must, be replenished periodically, there _is_ a limit to their life: At some point, the fabric will no longer take a DWR (Durable Water repellent) coating anymore.

Reflectors

In the dark, or under dark, rainy clouds, I consider a reflective vest to be very helpful.
No, it does not guarantee visibility, because it depends upon:
1. light being available to strike the reflective material on my torso;
2. and for that light to reflect in the direction of the object I am fearing–namely, cars and trucks.
Even then, the hope is that the driver is not texting, or paying attention to some display-screen!

In the dark, I try to wear reflectors on my wrists/elbows as well. The ones available in shops are too tight, restricting the essential flow if air which cools down my forearms; so, I’ve made my own.

They can be of use on ankles. But, as with wrists, they restrict the flow of air into my pant legs. As of this writing, I’ve not made my own, but then I don’t consider them to be as crucial as reflectors on the torso and wrists. In warm weather, I do carry them as a spare (to use just in case I don’t return before dusk), because they are compact and high-ROI as a spare.

Lights

I’ve discussed this elsewhere. I use one on the handlebar, pointing forwards for legal compliance, and one pointing backwards, at my torso.
Do I use lights in daytime? No. It’s rarely gotten dark enough, even under the stormiest clouds! A reflective vest is my preference; when commuting in rain, I often wear one.

Are lights useful at night? Not the way they are legally mandated–which is to light the path ahead! Cycling laws are frequently outdated and carcentric. Most cyclists, in most situations, purchase lights to protect them against cars; for that, the usage promoted by products often falls short–to put it politely.

Picture a driver, on a rainy night in a city. You are dealing with a person inside a box, with less than full command/awareness of his/her/per surroundings (because per is inside a box, and does not have 360-degree vision, or full knowledge of what may be approaching the perimeter of that box), in an environment too tight for manoeuvring a living-room-on-wheels, with no eye-contact possible with the person within, surrounded by glass upon which every single droplet of rain looks like a bicycle light, on urban roads with signage and traffic-control for which cyclists are an after-thought.
And per approaches an intersection that you are approaching from a 90-degree angle, with your light pointing forwards. Per’s attention is, at best, focused on threats–which, as with you, means other cars! Even pedestrians get more attention!

This has led me to use a light that points at my torso–the object that I am trying to protect. Not the bike, not the path ahead, but my life and limbs.
I want to let that driver know that there is a human on per path.

I wear a reflective vest as well. But, in practice, the amplification I hope-for might not always be occurring, considering the driver’s angle of approach, or the folds in my clothing as I lean on the handlebar, or the unzipping I do to tweak my temperature.

Bike Lights’ Brightness

As outlined above, outdated, carcentric laws mandate a usage of lights that is dysfunctional for what cyclists purchase them for. This has resulted in cyclists purchasing lights of increasing power, in the vain hope that the human rider will become more visible to approaching cars, even though the light is, most of the time, pointing in a wrong direction!
Then, I and other cyclists end up on two-way bike paths, blinding each other with the same!