I have noticed a dripping sound, and it appears to be on the flat roof above the bathroom ceiling. The bathroom is above the kitchen. When the dishwasher is on it's 3rd and final flush the dripping starts and finishes about 5 minutes later. Additionally, I am hearing a dripping somewhere behind the second half of the upstairs bathroom (not same area) when the hot water has run for a bit. This is a vanity, and large mirror attached to the wall, the sound is somewhere behind, and ends after a few minutes Any suggestions, or ideas of what could be causing this would be helpful. I know the dishwasher issue must be something outside. The bigger worry is the one in the wall, very faint but there.

Sounds like expansion of the piping due to hot water. If you only hear with hot and not cold, it is expansion. Very hard to "fix " without opening the walls.

I think Brian nailed it. Knocking, popping, creaking and crackling pipes are a common complaint, especially in colder weather. I can tell you what's happening and take the mystery out of it but you're not going to like the repair. When a draw is made on a hot water line the line expands against a pipe strap next to a stud and when you drain hot water from your tub or shower it goes into a chase that is a little cooler then room temperature. The heat expands the pipe causing it to rub against the stud to which it is pipe strapped. This is the sound you hear. As it cools it contracts and the noise is heard again. To repair it you must tear open the walls and locate the pipe strap that's causing the problem and shim it tight. Most people when they learn what causes it just elect to live with it. Good luck and thank you for rating my answer. Tom