There is not a lot of information on the internet, primarily because gill netting is illegal for most people in most of the Lower 48.
The following information was collected on some lakes in Minnesota, primarily in the fertile, southern part of the state.
One problem with this data is that a perch caught in a 19mm (3/4") mesh trap is about the size of a toothpick. What would the data look like if it were adjusted for fish weight?
32mm (1-1/4") mesh monofilament caught the greatest weight of bullheads but the 38mm (1-1/2") mesh caught 80% as much mass and you would have fewer to clean.
51mm (2") mesh, monofilament caught the greatest mass of carp, white suckers and bluegills. The 38mm (1-1/2") mesh caught about 80% as much mass in bluegills as the 51mm mesh and might be a good compromise if you were targeting both bullheads and bluegills.
The greatest mass of yellow perch were caught with the 19mm mesh nets.
|Same picture as above but reposted for convenience.
The mass for bluegills, on the other hand, shows very little gain from 38mm to 51mm suggesting that the optimum is somewhere in that region.
If the only fish in your ponds are bullheads, you cannot go wrong with a 32mm or 38mm mesh monofilament gill net.
If the water you are fishing has significant numbers of suckers and carp, then you cannot go wrong with a 51mm mesh, monofilament supplemented with a 32mm or 38mm net for bullheads.
As always, you are responsible for complying with your local laws.