Ponding water
Water that sits on a flat roof for days usually means the falls aren't right or the deck has sagged. Over time it speeds up wear and can find its way in.
Blisters, cracks and splits
Older felt roofs often blister and crack as they age. Small areas can sometimes be patched, but widespread cracking usually means the covering is at the end of its life.
Repair or replace?
If the deck underneath is sound and the damage is local, a repair may be fine. If the deck is wet or rotten, replacement with a system like EPDM rubber or GRP fibreglass is usually the long-term fix.