Gimme a sec to reach for my big can opener to open this industrial-size can of worms...

Let's start simple. All bikes (except experimental ones probing this subject) have trail/caster; Caster is the perpindicular distance from the steering axis to the center of the tire/road contact point. Trail is the similar measurement from the steering axis/road intersection to the center of the tire/road contact point, parallel with the road. Both pretty much mean the same thing. Henceforth I will mostly use the term trail, except when talking about caster angle on cars, which is analogous to the steering axis (head tube) angle on bikes.

The reason for trail is to have some centering effect on the steering when the bike is moving, i.e., stability, even so much as to be able to ride no-handed. With most bike geometry, having the fork straight and inline with the steering axis, results in too much trail, this can make the steering feel heavy, and also wheel-flop, where, at rest, weight on the bike causes the steering to flop one way or the other, unstable. So, the fork is either curved forward, or kept straight but inclined forward, to reduce the trail to a reasonable amount, which on most bikes is about 40-70mm.

A longer trail provides more steering centering force at speed, generally a good idea, unless excessive. Let me digress for a moment:

The Chevy Monte Carlo, I think second generation so 1973 I believe, GM's goal was for it to behave on the highway, especially high speed interstates, like Mercedes-Benz sedans, designed for autobahn speeds. 'Benzes had a lot of caster angle (for a car) so long trail. GM copied this caster spec for the Monte Carlo. And, the Monte's larger tire diameter meant that for the same caster angle, the trail value was longer, so a longer moment arm from the steering axis, that was the first mistake. The second problem was, the steered mass of the suspension (knuckles, brakes, wheels, tires, steering linkage) of the Monte weighed a ton compared to the carefully designed and svelte 'Benz components. Third, the front-end mass of the whole car, was much heavier than the 'Benz, so once moving, acts as an additional forcing imput. A freeway speeds, you turned the wheel a bit on the 'Benz and it recentered itself positively, with little to no overshoot. The same move on the Monte (as originally designed), hands off the wheel, the heavier steered mass and longer moment arm and heavier car mass, gave the correction more momentum, and it overshot center and went the opposite way, further than original steering input, then went the other way in an even greater overshoot, etc, and you had "divergent oscillation", the car, um, "departing from controlled flight", to borrow an aviation term. At this point the design was fairly sealed, expensive tooling was already complete, so no time to redesign, so GM added a damper, essentially a shock absorber with equal forces in compression and extension, mounted horizontally under the front end and connected to the lateral steering linkage, and this was sufficient to "critically damp" the oscillation to achieve steering centering without overshoot. So regarding that copied caster value, to quote my father, "A little knowledge is a dangerous thing."

This problem became further complicated with the debut of front-wheel-drive cars, so under engine load, the forces on the steering could get reversed, the tire contact patch, behind the steering axis, but being pulled *forward*, unstable, just like backing up a car or bicycle with positive trail. And, they hadn't mastered torque-steer, caused by "spindle offset", the horizontal distance from wheel center at wheel spindle height, to the steering axis. So the same thing happened on cars with much smaller and lighter steered mass and smaller trail. Some added steering dampers, some solved with better suspension design to eliminate spindle offset.

How're we doing? Are you with me so far?

Getting back to bicycles...

So... steering oscillation on a bicycle is a complex matter, involving trail, steered mass and polar moment (not just mass but how far the mass is offset from the steering axis), and, an additional complicator, that the bike and steered wheel can tilt left and right much more than on a car, so additional forces like camber thrust come into play, as well as the tilting inertia of the bike, cargo, and rider, and the rigidity/elasticity of the rider with the bike.

Trail can be a good thing, a stabilizing effect, if not excessive.

Steered mass and polar moment can be a good thing, a stabilizing effect, if not excessive. Notably, small-wheel bikes, less gyroscopic inertia due to smaller wheels so less steering stability due to that (more "agile/twitchy" steering depending on your personal preferance), can benefit from an increase in steered mass and polar moment, to help "calm" the steering, this is known as a "mass damper", which exist in many forms. The location of that mass, forward or behind the steering axis, also matters a lot.

(to be continued, I want to post the above before lost)

It's very late, I'm tired. I will try to continue this tomorrow or the next day.