No formula is known. Since Latin squares and Steiner triple systems give strongly regular graphs, lower bounds on the numbers of these structure give lower bounds if $n$ is square or if $n=v(v-1)/6$ and $v\equiv1,3$ mod 6. (Presumably these lower bounds are very weak.)
According to Brouwer's tables we have exact enumeration up to 36; the numbers on 37 and 41
are not known. (I expect that the number of srgs on $p$ vertices, $p$ prime increases with $p$, but this has not been proved.)

Of course we do not know the number of isomorphism classes of graphs on $n$ vertices. We have a procedure that allows us to compute the number for moderate values of $n$, and we know that asymptotically the number is $2^{n(n-1)/2}/n!$. Given this, our knowledge for strongly regular graphs does not seem quite so bad.