Drip / click sounds in basement ceiling

Probably normal dripping inside plastic based waste plumbing.

Frost has built up inside the sewer vent due to recent freezing exterior temperature. Use of the sink (& probably moreso the toilet) brings up some warmer & moister air when it vents the sewer pipe. The inside of said vent is frost moreso than ice, which is a fair insulator at freezing point which is pretty much where it is.

As the new charge of air is cooled to the freezing point, it melts some of the frost and loses some more water as well. The water runs down to the trap without making any noise, but the overflow from the other (sewer) side of the trap might drip for a few minutes. Because of how sound conducts through solid piping different from liquid-filled or gas-filled sections and/or liquid-to-gas surfaces, it is very difficult to localize the source of the sound.

If you have an inspection port in the basement, try blowing some hot air with a hair dryer. *** Constantly assume imminent egress of filthy-smelling slime. ***

IDK, but that's some strange construction:

I can hear some dripping sounds in the basement when I use the sink in the toilet on first floor. On top of the basement we have a kitchen (below the toilet).

So, between the toilet on the first floor and the basement, somehow you've have a kitchen. Must be what, about 4" high? Where did you find the appliances to fit?

I ripped out my basement ceiling looking for a drip. It was my wife in a rocking chair upstairs.

He's not in the US. In much of the world it goes basement, ground floor, first floor. First floor is what we in the US call second floor.

My guess, hot water in a galvanized waste pipe makes ticking noises with thermal expansion.