Exact formula for the number of spanning trees of a rectangle
According to https://www.cse.ust.hk/~golin/pubs/ANALCO_05.pdf there is no closed-form formula known.

According to http://arxiv.org/pdf/cond-mat/0004341v1.pdf the number is asymptotic (for $n$ and $m$ both large) to
$$\exp (z_{\mathrm{sq}}mn)$$
where
$$z_{\mathrm{sq}}=\frac{4}{\pi}\sum_{i=0}^\infty\frac{(-1)^i}{(2i+1)^2}\approx 1.16624$$
but I'm not sure whether this is a rigorous bound or the result of heuristic physics-based reasoning. The same paper also gives asymptotic formulas of similar type when $m$ is fixed to a small constant and $n$ is large.