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What are the grammatical rules determining whether to use which or what?
I usually go by intuition. It's "What car are you looking at?" and "What bicycle do you like most?" but "Which song did you sing?" Quite often, I don't know whether to use what or which.
How can I tell?
Which is used when there is a selection of choices, and you are picking one.
What is used for a description, or if you don't know what the choices are.
For your examples, it should be:
Which car are you looking at?
This is because you are probably next to your friend and have an idea of the selection of choices.
Which bicycle do you like most?
I'm assuming here that you are both in a store or looking at the same webpage, so that you know your choices.
What song did you just sing?
Here, you don't know the choices that were picked from, so you use what
There really isn't a whole lot of difference. Although which may be slightly more formal, what may be used without occasioning discomfort.
That said, which does seem to narrow the range of choices somewhat. If I say
What car do you drive?
that leaves the possible answers open to the set of all cars, whereas
Which car do you drive?
may feel like a question you would ask about a finite set of automobiles, the choices being limited to, say, your car or your wife's.
But this is not a hard-and-fast rule.
From this answer at EL&U.SE (and other answers to the same question):
For example, contrast which one do you want to eat? (when selecting from a menu) and what do you want to eat? (when you want suggestions to make a menu).
In this case, which is not a good replacement for what, and the opposite replacement isn't very good either.
