Below a certain speed, a trace behaves like a piece of wire: resistance, a bit of inductance, done. Above that speed, the trace's geometry (its width, its height above the ground plane, the dielectric between them) starts to matter as much as the copper itself. The rule of thumb: if the signal's rise time is short compared to the time it takes to travel the length of the trace, you're in transmission-line territory whether you planned for it or not.
Reflections are the tell
A trace with a controlled impedance that doesn't match the driver or the receiver at either end will reflect part of the signal back toward its source. On a scope, this shows up as ringing: overshoot, undershoot, a signal that looks like it's fighting itself right after every edge. It's not noise from the environment. It's the signal talking to its own reflection.
This is why a fast single-ended signal often wants a 50Ω trace, and a differential pair wants a controlled differential impedance, usually 90Ω or 100Ω depending on the standard. Those numbers aren't arbitrary. They're chosen to match the driver and receiver hardware upstream and downstream of your board.
You can calculate it, but you should still measure it
Field solvers and stack-up calculators get you close, using trace width, copper weight, and dielectric height and constant. But real boards have manufacturing tolerance on every one of those numbers. A ±10% swing in dielectric height alone can shift a nominal 50Ω trace by several ohms. For anything genuinely speed-critical, a controlled-impedance fabrication spec and a coupon measurement from the fab are worth the extra line item.
For everything else (most digital I/O, most sensor buses), this is a background fact worth knowing, not a rule you need to apply to every trace on every board. Know where your rise times actually put you, and spend the controlled-impedance budget only where the signal needs it.