What screen to use to keep flies and gnats out at night?
Bugs come in a lot of sizes, so unless you measure the bodies of the ones that get through in your area, it's hard to give you a specific size.  In general, though, screen fabric comes in a few nominal meshes intended to handle the typical-sized bugs.

I would start by just seeing if your screens are coarser than the "fine" mesh sold in your area.  Measure the holes or count the "threads per inch".  Compare that to the finest mesh they sell at the local store.  If you use threads per inch, be sure to compare against similar weight screening because different diameter wire or filament will have different gaps between them for the same count per inch.

If the finest they sell has a substantially smaller opening, I would just assume that it's fine enough. If you already have what's considered a fine mesh, you may need to do some research, like measuring bug bodies, or finding out what insect the local pest actually is and how small they are.

If you need openings smaller than the fine mesh sold in your area, you can get screen fabric online that goes down to almost microscopic size.  Some very fine mesh fabric is available for window screens.  The super-fine mesh is typically used for purposes other than keeping bugs out of your house, and can get expensive.