Which nonabelian finite simple groups contain $PSL(2,q)$ for some $q$?
Obviously $PSL(2,q)$ themselves do. Also, as $PSL(2,4)\cong PSL(2,5)\cong A_5\subset A_n,\; n\geq 5$, alternating (nonabelian simple) groups do as well. I believe I have read somewhere that $PSL(2,q)$ embeds in $PSp(2,q^2)$ as well, though I can't remember the reference so I wouldn't put money on it.
Of the finite simple groups listed by the classification theorem, which others contain a $PSL(2,q), \; q\geq 4$?
I would appreciate a reference since it would be unreasonable to ask for proofs.
ADDENDUM: In response to Derek Holt's comments, let me clarify that I do not need to know which $PSL(2,q)$'s are contained in which simple groups. My purpose is this: I am trying to prove a theorem about simple groups. I have the result for $PSL(2,q)$, for $q\geq 4$, and the family of groups for which it holds is upward closed. This deals with a lot of simple groups (for example all alternating groups, per above), and I am trying to figure out which simple groups I still have to worry about.
I have now convinced myself that all finite nonabelian simple groups apart from ${\rm PSL}(3,3)$ and the Suzuki groups ${\rm Sz}(2^{2n+1})$ contain ${\rm PSL}(2,q)$ for some $q \ge 4$.
I don't feel like writing a detailed proof. It might help you to look at the list of the minimal simple groups (i.e. simple groups with no nonabelian simple group as a proper subgroup) although I guess you would still have to worry about the possibility of a group only having one of the above exceptions as simple subgroups.