Recently, my bike often develops a front wheel wobble when I ride hands-free at slow to moderate speeds. The wobble quickly deteriorates, to where I think I'd go down if I didn't return my hands to the bar. It doesn't always happen - but often enough.
Wondering if the cause for the wobble was an issue with the front wheel, I checked for trueness. I've had my bike and its wheels for a little over a year and have put about 5000 miles on it, so some imperfection might be expected - but I'm not sure how to judge the severity of out-of-trueness. I detected a very slight wobble, with the rim moving away from my brake shoe by maybe 1 mm at the farthest point, where it was just barely touching at the closest point. Is this enough to be worth correcting, or should I look elsewhere for the source of my front oscillation?

Try putting your knee against the top tube. Read more: https://sheldonbrown.com/brandt/shimmy.html

If your lateral true is only out 1mm (which is more/less considered the allowable standard) I doubt this is the cause of your wobble. Many bikes will exhibit the hands-off wobble you mention and it is something you can just live with by not taking your hands off the bars for long. (All two-wheel vehicle front ends have a natural resonant frequency which manifests itself in a diverging wobble.) Older, less-rigid, more-worn bikes seem more susceptible in my experience.

I disagree with the cited reference that says bearings have nothing to do with the problem. While it is possible to have perfect alignment and bearings and have a wobble, it is also possible that a wobble is much worse when bearings are loose. One thing to certainly check is to be sure your steering head and front wheel bearings are up to snuff. Any play there will aggravate the problem in my experience and I have had bicycles with a wicked wobble that were more/less cured with a properly adjusted headset. And wobble or no wobble, you want these things to be adjusted and working properly regardless.

- Mark

I'd be more concerned with the wobble from worn or improperly adjusted bearings front or rear. But as far as trueness, never really had one that caused an oscillation. For as little as it costs and how long it takes, Just get them trued. For the most part, they stay true, unless you constantly are jumping and bending rims. It's almost like the steering alignment with a car. Do it once, it stays aligned unless ball joints or whatever else wear out and are replaced. The worst I had a bike wheel untrue, it has that roll that just feels like there's a lump in one spot. 1/16 to 1/8 inch bothers me and that's with the bike turned upside down that it's most noticeable, but that's still rideable, but I will get them trued. Never regretted getting a wheel laterally & radially trued. My biggest worries there are what it may do to prematurely wear a hub & bearings out.

the worst case of wobble i've ever had, where it was unridable at 10mph when handsfree, was when, as an experiment, i switched wheels from 630 (27") to 559 (MTB). on the 630's the ride was dead solid at any speed.

on another bike, when loaded for touring with 559 wheels, it was very bad handsfree, starting at about 20mph. without any touring baggage it was fine. recently, i switched to 650c wheels and the wobble came back.

my experiences lead me to believe that front end weight, or rotating weight, and the flexibility of the frame, or lack of it, are significant factors. so i suppose some experimenting with different weight tires/forks may make some difference.

BTW, as you can see, my problems didn't have anything to do with bearings or loose spokes or out of true wheels.

Wobble cause could be high back wheel load in conjunction with low front wheel load. Try putting something heavier (aero bars worked for me, even bull horns) on the front (bars) and see if wobble disappears.

On my bike it used to happen when I put lots of things on the back rack. Not anymore, after mounting aero bars on top of bike's flat bar.

I appreciate the thoughts, and have a few things to check. I'll see if shifting weight around on the bike makes a difference, or the knee-stabilizing thing. Will also check headset bearings. Is that just a matter of checking for play, or is there more to it?

Stand next to bike, apply front brake tightly, try to push bike
forward and backward with the brake locked on.

If there is play, you will feel it. Tighten the headset bearings
(according to method for headset you have), and lock everything
down when the play is gone. I usually tighten these to the point
of a slight preload......you will get varying advice in this regard.

My guess is that is in the headset or maybe wheel bearings. If it is a wheel out of true enough to cause a wobble you would likely notice with your hands on the handlebars and at nearly all speeds. Basically a significantly out of true wheel would be noticeable the whole time your riding and not just under certain circumtances.

shimmy is not well understood. There are a couple of good articles on wikipedia on it, here is a link to one It is the same phenomena found with grocery cart front wheels when you go too fast if that tells you anything.

I think it's clear that certain kinds of misalignment cause shimmy to be more or less likely to start. It wouldn't surprise me if an out of true front wheel would promote shimmy. However, the main phenomenon is determined by bike geometry and tire characteristics

And 300 gram bullhorns should sort it out.

If the wheel rim spins true or very close to perfection (<2mm lateral wobble), that ain't a problem. Then check the hub bearings. If they spin smoothly without lateral knock or wobble, then check the head tube bearings and fork for problems. If these check out OK, check your front tire. You'd be surprised how much lumpy, out of round, and otherwise out-of-spec tires can screw up the ride of a bike that's otherwise in allignment.

If all the above (including the tires) are OK, the main frame could be out of allignment.