The only examples of $\sharp P_1$ complete problems I've seen are fairly abstract : e.g. here https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/enumerate.pdf
Valiant proves that there exists a $\sharp P_1$ complete sub-pattern matching type-problem, but he doesn't give it explicitly.
I know it was conjectured ( for example, it's mentioned here: https://www.sciencedirect.com/science/article/pii/S030439750300080X ) that the number of self avoiding walks of length $n$ might be a $\sharp P_1$ complete problem. Has there been any progress on this problem? Or are there similar counting problems (some explicit combinatorial family) which have been shown to be $\sharp P_1$ complete?