I’ve experimented a lot with attempts to make shoes waterproof, snowproof and sandproof (this is an issue in desert ultras).
My conclusions:
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There are plenty of trail shoes that advertise as “waterproof”. They are generally a better solution than waterproof covers for regular shoes.
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They are still crap. In most circumstances, I’d go with Fleck’s answer.
The problem is firstly it is hard to keep water out. At the minimum it tends to run down your legs and into the shoes.
The next problem is that Goretex & similar are very good at keeping water in. Your feet will get wet, and continue getting wetter the longer you run. There is no drainage at all with these shoes.
A related problem is that sweat accumulates in waterproof shoes. You may not think this is a problem when running in the rain, but you might be surprised.
Goretex works OK to release “water vapour” in applications like jackets where there is a layer of air between you and the garment. It doesn’t work that way with tight shoes.
Short run, cold day, light rain, no deep puddles to navigate; waterproof can be a good choice. Otherwise, like Fleck says, let the shoes get wet, rely on good socks.