Just like many other structural features in your house, roof should be a part of routine inspection and maintenance. Depending on the type of roof you have (wood, tile or concrete), the warning signs of damage may be different for each case.
Below are common signs of roof damage to look out for, and avoid small problems from turning into serious ones:

Bend or bulge in roof that occurs mostly because the roof has not been properly installed. This may require replacing the roof.

When shingles start getting loose or fall off, that is also a sign that you may need to replace the entire roof.

Discolored or tainted shingles need to be repaired.

Any type of water damage or leaking through the ceiling needs quick attention.
Check for presence of termites or mold. Termites travel through mud tunnels so if you see tunnels, your house is at a risk of termite infestation. Molds appear as greenish stains on roof exteriors. Water problem can lead to mold or bacterial growth under the shingles. Both termites and molds need to be treated right away.
Extreme weather conditions, such as snow, hailstorms can also damage the roof. Look for any bruises to shingles.
Cedar roofs need to be replaced every 20 years. If roof is made of tiles, then look for any breaks in individual tiles. Concrete roofs do not need to be replaced but should still be inspected to check for cracks or stains.