Are free groups residually simple? Let $F_k$ be a free group of rank $k\ge2$. Given $g\in F_k$ with $g\neq e$, is there $N \vartriangleleft F_k$ such that $g\notin N$ and $F/N$ is finite simple?
 A: Not the simplest approach, but with a strong conclusion: it's been proved on the one hand by Margulis and Soifer in the 80's that, for any $k,d\ge 2$ $F_k$ can be embedded as a Zariski-dense subgroup of $\mathrm{SL}_d(\mathbf{Z})$. And on the other hand (earlier) by Weisfeiler that every Zariski-dense subgroup of $\mathrm{SL}_d(\mathbf{Z})$ has the property that its image modulo every large enough prime is surjective. Combining, we reach the conclusion that $F_k$ is residually-$P_d$ (with surjective maps), for $P_d$ the class of $ \mathrm{PSL}_d(\mathbf{Z}/p\mathbf{Z})$ where $p$ ranges among primes (or even among any infinite set of primes).
It sounds to me plausible that $F_k$ is residually-P for any subclass $P$ of the class of nonabelian finite simple groups with unbounded order; I'm not sure what the current state of art is (the above uses results of the 80's but there has been much progress since then).

Edit After finding Derek's suggested reference: 
Then it seems that the first proved case, which is enough to answer the OP's question, was the family $\mathrm{PSL}(2,p)$, $p$ prime: it is due to Peluso,  Comm. Pure Appl. Math. 1966. 
Next Katz and W. Magnus (Comm. Pure Appl. Math. 22 (1968), 1-13) proved it for the family of all alternating groups, and also extended Peluso's result.  The case of arbitrary families of alternating groups is due to J. Wiegold (Arch. Math. (Basel) 28 (1977), no. 4, 337–339). Lubotzky (Proc AMS) made the connection to the then recent results of Weisfeiler to considerably enlarge the class of known examples among families of groups of Lie type (which was extended by S. Pride in the 70's).
The general case (arbitrary families of nonabelian finite simple groups) is indeed due to T. Weigel (1992,1993) in papers Residual properties of free groups, I,II,III in J. Algebra, Comm. Algebra, Israel J. Math.
