Generally speaking, a bicycle with a longer wheelbase will be more stable when ridden slowly. Those designed for quick maneuvers are generally very short (criterium) wheelbase designs. But the tradeoff is, of course, that the long wheelbase makes those quick direction changes a bit more difficult.
Modern bikes tend to have straight forks and very steep head angles (where the fork attaches to the frame). Both of these shorten the wheelbase and make the bike more "responsive" aka "twitchy." Shallow angles and curved forks go the other way. Stability and comfort.
I hate to recommend anyone get a heavy bike, but there is some effect. The lighter the bicycle, the higher the center of gravity overall (i.e. you).
On the other hand, track riders can ride slowly, even stand in place. They do this with skill and the lack of a freewheel. You can even ride backwards on a track bike. These bikes are also twitchy in the extreme. But it is rider skill. With enough practice you can learn to "stand on the pedals". That is, balance the bike in place, though with a freewheel you need to also use the brakes You've probably seen people doing this when stopped at a light, balanced upright.