Wobbling harleys - 99.9% riders fault I say... (either by stupid decision or poor maintenance) - no idea if this is a specific issue with newer (post 90's HD's) but nevertheless...

Wont happen to my old lady - steady as - but should I ever come across/experience the feel, first thing I do would be to pull back on the throttle and slow down to regain control - not continue as if nothing is going on
Then(!), start asking myself wtf just happened and try to work my way back from there.

Poor (or too soft) rear suspension, one cause (along with overloading it resulting in pretty much the same) (just one of several common problems) - (wrong tire pressure another common problem) - not the bike to blame - its the rider making a bad decision @ the end of the day

Old news this one........it was solved in 2010 with the new frames. Pre 2010 the frame was unable to handle the weight and load placed on it causing it to flex. Plenty of accesory mobs made kits to stop them flexing and still do today.

all rubber mount dyne's can wobble, sort of at high speed as swingers in gear box, new rubber sporty same. they can allow motor and gb to move forward to rearward just a bit, but can fit a gizmo to stop that. I broke a rear engine mount on a twin cam dyna I had and when I hit 160kph at drags she went into a wobble, I opened her up harder to try to pull her out of it and she got worse so I used rear brake while keeping throttle open that pulled her out of it. cow at l a cycles sorted it out for me and never did it again handled better too.

Amazing how a piece like this brings out the same Harley V Japanese too and fro in the comments. Humorous.