Set Nudge Value to 1 ms ⌃⌥1
Quickly change the Nudge value to one millisecond. Realistically my smallest resolution for nudging might be SMPTE frames or quarter-frames. The difference of a millisecond is difficult to discern without multiple scales to compare it to (audio and video).
You can move regions to a different point in the same track, or move them to another track of the same type. You can move an audio region to another audio track, for example, but not to a software instrument track, or vice versa. You can also move regions between open Tracks area windows, or between different open projects.