Generally speaking, how much runout can a bicycle wheel have and still be considered true?
If your wheel is trued (lateral movement) to within 0.5mm it is considered good. Same 0.5mm tolerance for the roundness of the wheel. Just remember that you may need to take the "stress" out of the wheel as the spokes can wind up with the torque applied to the nipple. It is that pinging sound that happens when a freshly built wheel is ridden for first time. Simply squeeze the spokes together, as though you were playing it like a harp. You should hear some pings then check if it is true and round, adjust as needed.
You mentioned it was your front wheel so you can flip it round to check if you have it dished equally (lateral offset) as is the norm for most front wheels. A blob of blutac and a pin on the fork blade can be used as a guide. And finally the wheel will go out of true with use especially during the first few rides, so check it after a couple of months.
Hope this helps.
The standard for what "true" means varies by mechanic and the intended use of the wheel.
A perfectly true wheel, in the mechanical sense, may not be attainable. Roundness and lateral runout are measurable to .001 millimeters using commonly available shop tools. Most mechanics do not consider that tolerance necessary, but as that is the limit of the tools, that is what I would consider perfectly true. However it requires 4 things:
- Perfect roundness, measured by placing a gauge under the edge of the rim, and spinning the wheel.
- Perfect lateral runout, or rather a lack of runout. This is measured by placing a gauge at the brake track of the wheel, and rotating the wheel to track the runout.
- Perfect Dish. Dish refers to the centering of the rim on the axle, or in some cases between the dropouts of the frame. This is measured using a truing stand or a dishing tool.
- Perfectly even and balanced spoke tension. Measure using a tensiometer, the value here will vary significantly based on the type of spoke, the size of the rim, the material of the spoke nipple. But the constant here is that all the spokes should have exactly equal tension.
I am well aware that the description here is of a mechanically perfect wheel, and that such a thing does not exist. It would require a perfectly formed rim, a perfectly machined hub, which perfectly drilled and finished spoke holes, and that each spoke be drawn from a perfectly pure metal at exactly equal and even dimensions.
The beauty and simplicity of the bicycle wheel is that while our manufacturing tolerances are not up to that standard, the design of a spoked wheel allow you to offset the tolerances in one area with spoke tension. If a rim is not perfectly true and round, more or less spoke tension can be used to pull it true and round.
So a "perfectly" true wheel might not exist, but a functionally true wheel is relatively easy. And "perfectly" true isn't necessary.
Wheelbuilding machines can be configured for a .1mm precision. But the choice of tolerance is up to the wheel factory. Best machine-built wheels are trued to this maximum of .1mm precision, which is pretty good. Most man-built wheels have a run-out around .5 mm when new.
But when you are building your own wheel, or you are fixing it, a tolerance of around 1 mm is acceptable imho. More than 2mm and you will have problems with brake pads or you may feel some shaking while riding.