Links to some open problem collections:
Erdõs Open Problem
Open Problem Garden
Graph theory and Combinatorics (West)
Computational Geometry (Demaine, Mitchell, O'Rourke)
My questions on MathOverflow
And a random collection of less famous problems that I've encountered sometime and liked enough to post about them.
12. Can facility location be better for second player against powerful first player?
11. Can lazy cops catch a robber on a planar graph?
10. Is there a perfect matching in every F or its complement?
9. Can every integer matrix be made even with swaps?
8. Fit boxes in cube
7. Covering inequality for sets of intervals
6. Half red half blue AP
5. Sylver coinage
4. Unavoidable patterns in words
3. Good colorings in randomly-colored graphs
2. Coloring Zd without large mono
1. Discrete Lebesgue density