Here are some of the more commonly known problems that are PSPACE-complete when expressed as decision problems. This list is in no way comprehensive.
Games and puzzles Generalized versions of:
Logic
Lambda calculus Type inhabitation problem for simply typed lambda calculus
Automata and language theory
Circuit theory Integer circuit evaluation
Automata theory
Formal languages
Graph theory
Others
See also List of NP-complete problems
Notes
References Garey, M.R.; Johnson, D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: W.H. Freeman. ISBN 978-0-7167-1045-5. Eppstein's page on computational complexity of games The Complexity of Approximating PSPACE-complete problems for hierarchical specifications