On my last trailer, Skyline Koala 21CS, I would leave the stabilizer jacks down. I did it because with just wheels and tongue jack, it tended to wiggle a lot.
However, I've found that my current trailer, Jayco Jay Flight 23RB, seems to be much more stable than my previous one. I can walk around in it and it doesn't wiggle much. Not sure why the 23RB is that much more stable than the old 21CS. Because of this inherent stability, I do not put the stabilizer jacks down now.
Erroll and Mary Doss and Duffy (RIP)
2017 Jayco Jay Flight 23RB
2014 F150 SC, 4x4, HD Pkg, Sterling Gray