How tile roofs fail here
The tile is the sunshade. The underlayment beneath it is the waterproofing. After thirty or so summers of attic heat radiating up and sun beating down, the old asphalt felt turns brittle, cracks along the laps and lets water through at the nail holes. From the street the roof looks perfect; from the attic you can see daylight.
Add a few cracked tiles from a satellite installer's boots and the felt under them goes even faster.
Small repairs
A cracked or slipped tile gets lifted out, the felt under it checked and patched, and a matched tile slid back in. Broken bird stops and eave closures get replaced so wind-driven rain and rodents stay out. Loose ridge and hip caps get reset with a mechanical ridge system or a modern adhesive rather than the old mortar that crumbles in the wind.
I keep a stock of the common concrete profiles from the tracts around here so matching is usually possible.
Flashings on tile
Chimneys, skylights and wall junctions on a tile roof are flashed with pan and step metal under the tile, and that metal is where most tile leaks originate. Repairing it means lifting the surrounding tiles, replacing the metal properly lapped into the underlayment, and resetting. Sealant on top of the tile fixes nothing.
Pipe flashings on tile use a lead or malleable metal boot dressed over the tile profile; the cheap rubber kind does not last.
When a re-lay is the answer
If the underlayment is brittle across a whole slope, or the attic shows staining in several places, patching individual tiles just moves the leak. The fix is a re-lay: lift and stack all the tiles, strip the felt, replace any bad decking, install new synthetic underlayment with membrane in the valleys and at penetrations, and reset the same tiles with new flashings.
It costs a fraction of a new tile roof because the tile is reused, and the house looks the same when it is done.
What to check yourself
From the ground with binoculars: tiles out of line, ridge caps sitting crooked, and gaps at the eaves where closures have fallen out. From the attic with a flashlight on a sunny day: any pinpricks of light, any dark stains on the plywood, any insulation that looks matted.
Do not walk the roof; concrete tile cracks under a careless step and clay is worse.