Spring comes from crimped yarn, and water plus time is what lets it come back
Two facts about the material. A carpet's resilience lives in the crimp of its yarn — each tuft is a fibre that was heat-set into a spiral, and the spiral is what pushes back. A leg standing on it for a year flattens the spiral and closes the air gaps between fibres, and the patch goes dead. The backing underneath is a different problem entirely: fabric plus latex or bitumen, and once it has been pressed into a dish it stays dished, because there is no crimp in it to spring back.
That gives the two-second test. Press the dent with your palm and then with your fingertips. If the fibre comes up when you lift it a little, the yarn is there and only flattened — that is treatable. If you feel a hard, smooth, shallow dish with a defined rim, the backing has gone and no treatment will lift it; that dent gets a patch or stays. Everything else on this page follows from which of those two you are feeling.
And the treatment is counter-intuitive in the right way. Ice is the tool, because it gives the two things the fibre needs at once: water and time. As a cube melts it releases water slowly into the crushed tufts; the fibre absorbs it, swells, and its glass transition drops, which is a complicated way of saying the yarn becomes supple enough to unfold. The cold also stiffens it once it has unfolded, so it sets. Twelve to twenty-four hours later the crimp is back and the pile stands up. An iron delivers the same water as a violent burst of steam plus 150 degrees of heat, and heat is exactly what set the crimp flat in the first place.
Three faults, three answers
Fibre that lifts under your fingers is the ice job and it costs nothing. Move the furniture, lay three or four cubes per dent spread out rather than piled, and leave them. Blot the water with a towel when they have gone — do not rub, which mats the pile — then tease the tufts upright with your fingers or a soft brush, working in the direction the pile already lies. Then leave it alone for a day, which is the hard part: walking on it before it dries presses it straight back down.
A hard dish with a defined rim is the backing, and the honest answer is a patch. Cut a square of matching carpet from the offcut every new carpet leaves — check the wardrobe, the loft, the back of a cupboard — and match the pile direction before you cut, because a patch laid against the pile is visible from across the room. Carpet fitters call this a plug patch: a square cut with a steel rule and a sharp blade, held with double-sided carpet tape, with the pile direction aligned.
Fibre that crumbles or snaps is a carpet that has aged past treatment — usually polypropylene that has had sun, or a cheap carpet in a walkway. Pinch a tuft: if it breaks off instead of bending, there is nothing to lift. This is where you stop treating and start covering: a rug over the worn patch, furniture moved to a new arrangement, or replacement. The distinction matters because the ice method on brittle fibre makes a fuzzy hole where there was a dent.