Water coming over the edge
The clearest sign is water spilling over the front of the gutter during rain rather than travelling to the downpipe. Watch the roofline during a downpour and note where it happens.
Staining and growth
Dark or green streaks down the wall below the gutter show water has been running there for some time. Grass, weeds or thick moss along the gutter line means silt has built up enough for plants to root in it.
Signs inside the property
Damp patches on an upstairs wall, peeling paint or a musty smell near an external wall can all trace back to rainwater that is not draining away properly.