Asymptotic bounds on number of 3SAT formulas with unique solutions
It is dense, just as one would expect.

The number of uniquely satisfiable 3SAT formulas with $n$ variables and $n$ clauses is at least $2^n$, since for every possible assignment to the $n$ variables there is at least one 3SAT formula $\phi$ for which that is the satisfying assignment, and there are $2^n$ such assignments.  A 3SAT instance with $n$ variables and $n$ clauses can be represented as a string of length $\ell = 3n \lg n$.  Therefore, the number of such formulas is exponential in $\ell$.