Every flat roof is laid to a slight fall, because standing water is what destroys the covering. When one begins to pond, the covering is no longer the first problem: the deck or the firrings beneath it have sagged, and recovering the surface without correcting the fall produces a new roof that ponds in the same place.
Small flat roofs over a porch, a bay or a single storey extension are the ones most often neglected, since nobody can see them and they only announce themselves indoors.
What you can see without going up
Look from an upstairs window after rain. Water remaining on the surface more than a day or two after a shower is ponding, and the shape of the puddle shows where the deck has dropped. A clear line of silt or algae marks the edge of the pool even once it has dried.
Check the perimeter detail too. The upstand where the roof meets a wall should be carried well up and tucked in or covered by a flashing, and the edge trim should overhang so water drips clear rather than running back underneath.
Reading the covering
Mineral felt shows its age by blistering, cracking at the edges and losing grit. Blisters are trapped moisture; if one is soft and wet it has already failed. A single layer over a damp deck will blister within a season.
Older asphalt crazes into a fine pattern across the surface, which is normal, but deeper cracks that open in cold weather are not. Modern single ply and rubber membranes fail at the joints and at the upstands rather than across the middle, so those are where to look.
The loft and ceiling tell you more
Get at the underside where you can. Staining on the deck boards, a musty smell, soft spots underfoot and daylight at an edge all locate the fault far more precisely than looking at the top surface.
A damp patch on the ceiling below is almost never directly under the entry point, since water tracks along a joist before it drops. Follow it back uphill rather than opening the roof above the stain.
Patch or replace
One split in otherwise sound covering, away from a joint and with a firm deck beneath, is a reasonable repair. Several blisters, failure at the upstands, a covering past fifteen years old, or any softness in the deck means replacement.
The deck matters more than the covering. Laying a new membrane over wet boards buys two or three years, and the second job costs what the first one should have.
Drainage and the edges
A flat roof outlet is a small hole that blocks completely with one handful of leaves, and then the roof becomes a tank. Clear it whenever you clear the gutters, and check that the outlet sits at the lowest point rather than where it was convenient to fit.
Where a flat roof abuts a pitched one, make sure the pitched roof is not discharging a concentrated stream onto a single spot of the flat covering. That wears it out locally and quickly.