Do only certain air planes leave "paper-trails" across the sky, if so which ones do?

Wellll, they all create paper trails with air traffic control towers..... If you mean VAPOR trails, then it depends of the dew point, temperature, etc. I believe the FAA actually posts notices/data on a particular day as to where when an aircraft will create contrails. It's usually jetpropelled aircraft at higher altitude that make them.

Heh, only ones that left the lavatory doors open!

I respectfully invoke my 5th amendment rights as my answer may tend to incriminate myself