List of NP-hard problems, where there is active research in practical heuristics
MaxSAT - people actually care about this because SAT solvers are so well-developed that often the best route for your favorite NP optimization problem in practice is to reduce it to MaxSAT and then apply one of the well-known solvers. Check out the SAT competition for benchmarks etc.

Clique-finders get used in computational biology and combinatorics, and the heuristic algorithms are shockingly good, as I recall.

Vast portions of Operations Research is devoted to algorithms, including heuristic ones, for solving cases of integer or mixed-integer linear programming.