Large classes which contain LOGSPACE for which strict inclusions are unknown
This is a favorite question of mine.

Fortnow showed, in his paper "Time-Space Tradeoffs for Satisfiability", that $NL$ is properly contained in $\Sigma_{a(n)} P$, where $a(n)$ is any unbounded function. That is, nondeterministic logspace is properly contained in alternating polynomial time with $a(n)$ alternations. 

Showing that $NL$ is not in $\Sigma_k P$ for a fixed constant $k$ would imply that $NL \neq NP$. (To see this, consider the contrapositive.)

It is open whether $NL = P^{\#P}$. The last time I seriously attempted to prove this, it resulted in the paper "Time-Space Tradeoffs for Counting NP Solutions Modulo Integers". I was trying to find some simulation of every language in logspace that would take $n^k$ time for some fixed $k$ when one has access to an oracle for counting satisfying assignments to a given formula. (This would imply $LOGSPACE \neq P^{\#P}$.) My approach didn't work, but I ended up using the same approach to prove time-space lower bounds for solving $Mod_6 SAT$ and other related results.

Uniform-$TC^0$ is properly contained in $P^{\#P}$. The proof is in Allender, "The Permanent Requires Large Uniform Threshold Circuits". Any improvement on this separation is open. (For example, proving uniform-$NC^1 \neq P^{\#P}$ is open, and proving uniform-$TC^0 \neq NP$ is also open.)