Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathit{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, if $R$ is Frobenius, these are the same as the injective modules.)

Question 1: Why do we kill the projective modules? What did they do in particular to provoke our wrath?

Obviously, one motivation for the construction is that it happens to result in a triangulated category (and I think, if done correctly, a stable $\infty$-category). But on the one hand, there are other ways of extracting a stable $\infty$-category from $R$, such as taking the derived category. And on the other hand, surely there are other classes of modules we could have chosen to kill to get the same effect? For instance, why not kill all dualizable modules or something (assuming $R$ is a Hopf algebra)?

Question 2: What is the geometric interpretation of the stable module category?

Question 3: What is the relationship between the stable module category and the singularity category of $R$?