To find a puncture is the only part that requires any initiative or imagination. The rest is just running through the drills. It's important - and reassuring - to trace the source of the tube failure to make sure you don't have the same puncture twice.
Pull The Tube Out
Once the tyre is off the rim on one side you can pull the tube out. Reach into the tyre somewhere away from the valve and pinch and pull the tube out. Leave the valve in the rim.
Blow the tube up with the valve still in the rim. You can find a puncture by listening for the hiss of escaping air.
The smallest, slowest punctures are hardest to find and may demand you immerse the tube in water as a last resort, look for bubbles – they may be very slow. If you plan to patch the tube later enlarge a tiny hole with a small blade to make it easier to re-find. You can still identify where to look for the source of the puncture by holding the valve against the valve-hole. Check the area of the puncture.
Check The Tyre
If you still can’t find a puncture take the tyre off the rim and carefully run the pads of your fingers round the inside feeling for a sharp foreign object trapped in the tyre. If you mount the tyre label – or the pressure information if the tyre has no label – next to the valve you can still match the tyre to the tube and look again for the hole in two small areas of the tube. If you don’t know which way round the tube was you can find the puncture by looking in one of two areas – unless the hole is dead opposite the valve.
Remove The Source
When you find a puncture match the tube to the wheel and tyre and check the tyre in the area where the hole was. This is easier if you mount the tyre with its label – or pressure information if there’s no label. Check the tyre for grit, glass, thorns or nails. Check for holes where they may have been. The flint, thorn or glass that caused the puncture may have been blown out by the escaping air, or dislodged in the last few metres when you were running on a flat tyre. Pick out whatever caused the puncture. Tweezers or a pin may be useful.
You may find a puncture has been made by a spoke or a spoke hole. Check the rim tape covers all the holes. File down any high spokes. In an emergency you can pad the spokes with strong glossy paper or a scrap of tough cloth. If you have a spoke puncture on the inside of the tube there is no need to look for a foreign body in the tyre.
Even if you don’t plan to patch a failed tube, don’t throw the punctured tube away until you can replace it. Even if you have another spare the next puncture may not be patchable – for example a valve failure. A failed tube can be patched and is better than no tube at all. Roll it up – in a different style to the way your good spares are folded – and carry it until you can get a new spare.