If you measure the thickness (eg
jewellery tool), then rims seem to fail at about 0.6 to 0.7mm.
Most recent (last 5 years or so) rims have a wear indicator - either a groove that disappears as the surrounding material is worn away, or an internal cavity that shows up as a hole or slot. What the indicator is depends on the brand of rim.
As rims wear thin, tyre pressure will start to fold the flange outwards, so the apparent dish in the braking surface becomes pronounced.
Don't ignore any knocking or pulsing from the brakes. It could be the bulge where the rim has started to split going between the blocks. If this happens, let half the air out the tyre and ride directly home, slowly, without using the brake on that wheel.