Here are my solutions for the first few numbers. Some of these are not optimal (fewest operators):

1. 44/44
2. 4/4+4/4
3. (4+4+4)/4
4. 4*(4-4)+4
5. 4!/4-4/4 (not optimal)
6. (4+4)/4+4
7. 44/4-4
8. 4+4+4-4
9. 4+4+4/4
10 . 4+4+4-sqrt(4) (not optimal)
11. 4!/sqrt(4)-4/4 (not optimal)
12. 4!-4-4-4 (not optimal)
13. 4!/sqrt(4)+4/4 (not optimal)
14. 4+4+4+sqrt(4)
15. 4*4-4/4 (not optimal)
16. 4+4+4+4
17. 4*4+4/4
18. 4*4+4/sqrt(4) (better than the solution at Math Forum - Four 4's Puzzle)
19. 4!-4-4/4
20. 4!+4-4-4