I've never had a problem with standard bat-wing antenna looseness/wobble on the three trailers I have owned, and this is the first time that I have seen anyone ask the question on the multiple forums I have belonged to for years. However, if you accidentally left it up and had a close encounter with a tree limb... Note that I can't answer the question with respect to other types of antennas, such as satellite varieties.
Happy Camping! ///// Richard D.
2006 4x4 Ford 250 SD / 2007 Flagstaff 827 FLS
One very patient wife and one furry child who travels with us. Forty-two years of trailering and camping, and I still have a blast.