Well - there are tons in my mic box I would not use. First thing is simply volume. When you are learning this might not be a problem, but a trumpet, or any valve or extending tube instrument can be amazingly loud. So a mic with a pad can be very important. Some mics are also too sensitive to on-axis wind - so some of the usual condensers might make pretty awful noises. For me, it would be a large diaphragm condenser or even something simple like an SM57, or a Beta 57.
A worse problem is that many brass players cannot play standing still, so the bell waves all over the place, and as the instrument goes off-axis it gets thin and weedy very quickly. Live players often use clip on mics to ensure this doesn't happen, but this close in position isn't brilliant for out and out sound quality - just too close for a quality recording. Some players also have amazingly clanky playing technique and their
valves make a lot of noise, so keeping them still to find the optimum position is very hard. A hyper might be better for some players if they stand still, so it can be a little further away, but others might need being in closer. If they are good and do lots of high note splitting, then a condenser is pretty essential. Giving a trumpet player headphones can also be troublesome, as those unused to recording will tell you the tone is all wrong, but it's because they've never heard what you hear from the front - the sound they hear is totally different, so expect comments on the sound. Lots of brass players cannot play without music, so a stand needs squeezing in for them. If it's your own
studio - also be aware they let the damn things dribble onto your floor - so if the floor covering or surface is precious, tell them - because spit stains! You need to be able to cope with identifying timbres that are 'art' vs timbres that are bad technique and need repairing. Telling a player he sounds very raspy or breathy can be a compliment or an insult, so take care. For me, brass is one of the hardest sound sources to get right.