Wow, thanks for all the interesting tips and info so far!
So I went upstairs to the bathroom, turned on the sink faucet full blast, hot water. After about 30 seconds, I started to hear the familiar dripping sound about once a second.
I turned off the faucet and the length between drips started to grow further apart. The sound of the drips sounded the same to me. Admittedly though, I'm not sure what constitutes "slightly" different popping sounds.
I then thought that I would try turning on the faucet full blast with cold water. The drips started up again but at a different pitch, half as loud, and maybe twice as fast. Turning off the faucet once again causes the length of time between drips to grow until I don't hear it anymore.
Looking under the sink, there's plastic piping all the way into the wall for the drain and copper piping for the hot and cold water supply lines.
From what you've described above, it sounds like the popping sound due to expansion/contraction shouldn't be occurring continuously should it? It should only happen during the initial transition from the pipes being cold to hot and vice versa. But I'm hearing continuous "dripping" sounds.
Then again, I don't have much of a background in plumbing so please correct me if I'm wrong.