If it’s not obvious (no condensation visible on the surface) you’d need moisture detector to measure moisture presence inside the wall. You may be able to find an inexpensive one (analog type) in a local hardware store, just type “moisture meter” or “moisture detector” in Google search box.
Those stains might be related to an inadequate insulation inside the wall and above the apartment’s ceiling. Digital thermometer would help you determine that just like you can see on a picture in my article – Stains on walls
This article also explains how those stains are generated and if they might be related to mold. However, the only way to be 100% sure that something is or is not mold (and what kind if it is) is to perform laboratory tests of the surface material picked from an area in question.
Let me know if you have more questions, just click “comment” below.