PROBLEM: Nuts, you got a flat! But once the tire is off and the tube is out, you can't see anything that caused the puncture.
SOLUTION: This is worrisome because if something sharp is hidden in the tire tread, it'll quickly pop the new tube.
First, gingerly feel around the inner surface of the tire. Don't like prickly surprises? Then wipe with a rag. It should snag on something sticking through, but your fingers do a better job.
Find something? Get all of it out. You might need to dig with the screwdriver tip on your multitool.
Can't find anything? This is why you should mount tires with the label centered over the valve stem. (It looks pro, too.)
Pump air into the tube and listen for a hiss on its outside circumference. See where the hole is in relation to the valve, then look the same distance from the tire label. This pinpoints the suspect area. If a close inspection still doesn't find anything, you can safely assume that whatever popped the tube didn't stay in the tire.
If the hole is on the tube's inner circumference, it came from something the rim strip isn't covering. Find the flaw and correct it before installing the new tube.
If you find two little parallel slits, your flat was caused by riding into something hard enough to cut the tube between the rim and tire. This is called a pinch flat or snakebite flat. It's rare if you keep your tires inflated to their full recommended pressure.