One of my hw questions asks: show that the worst-case running time of quick-select on an n-element sequence is Big Omega(n^2). Isn't Big Omega supposed to be the best case scenario? Is the question asking what's the worst best case scenario?

Big Omega can be considered the lower boundary of your function. In other words: past a certain point, your function will always be greater than the function it's big omega to. Or said even another way. f(x) = BigOmega(g(x)) if for every x past a certain value f(x) >= g(x). Hope that helps.

You can use it for all 3 ways (best case, average, worst). You just have to specify which one you are doing it for.