Rarity: US Army Fifth Aviation Regiment “Catscratches”  
Applejack: US Marine Corps Electronic Attack Squadron (VMAQ) Six “Lightning”  
Pinkie Pie: US Navy Strike Fighter Squadron (VFA) Fifty Six “Partycrashers”  
Twilight Sparkle: US Navy Fleet Air Reconnaissance Squadron (VQ) Ten “Omniteers”  
Fluttershy: US Air Force Fifth Bomb Squadron “Grizzly Riders”  
Rainbow Dash: US Air Force Twentieth Fighter Squadron “Daring”
 
From my research, each of these patches would be relatively plausible for their respective services and aircraft types. The Air Force uses shields and scrolls and weird distorted text for everything. Navy and Marines seem to like round patches with ribbons. Army aviation uses lots of different designs, but square shields are common.
 
If you’re wondering, Fluttershy’s design is both a bear’s face and also pointy planes dropping pointy bombs on targets that look like lines. There’s five elements total to signify the squadron number.  
“Omniteer” is a word I made up from Latin roots that means someone who knows things, I guess.  
The lightning on AJ’s patch is the same color used in the zap apple episode.
 
Some are better than others.