Would it be in poor taste to ask my supervisor for a former coworker's schedule?

I don’t think so at all. Let them know now, so when they need to replace someone, they just have to replace your old hours.

It is a perfectly reasonable request.

I agree. Go for it. If you don’t say something about it, someone else may & then they’d get those hours. Get in there & ask. And good luck. :-)

Not at all. Just make sure the management already knows your coworker plans on leaving!

How will your supervisor know you want it if you don’t ask?

Definitely fine, as long as (as @Likeradar mentions) you’re not breaking any confidences.

If your coworker has already submitted their letter of resignation, you should submit your request for their schedule – in writing – to your supervisor. Make sure the letter is dated and signed by you.

Go for it!

I believe you get different(higher salary) for that,right?. If not,then you must tell your supervisor in honest way. (he/she could just test you for some reason).

It’s not crass at all. Ask for it now before someone else does.