I worked in consumer lighting for a while, and spent way too much time looking at the system dynamics of lamp & dimmer interactions for an induction lamp we were developing, so let's clear up some misconceptions and lay down some truths.
Caveat 1: my knowledge is about 6-8 years out of date, but I doubt anything has changed drastically since then.
Caveat 2: this was all in the US, so I don't know much about the situation in other countries.
Leading-edge vs. trailing-edge dimmers: Leading-edge ones are the simplest (hence why they're the most common) because they use triacs, which do require a continuous small hold current to stay on. Trailing-edge ones necessarily use either MOSFETs or IGBTs, and are more complicated as a result.
Leading-edge dimmers create a large dV/dt at turn-on, which creates EMI as well as transient-response ringing issues with the lamp's EMI filter, if you aren't careful to damp it properly and add RC snubbers in the right places. Trailing-edge dimmers do create a dI/dt transient at turn-off which means the stored energy in the wiring needs somewhere to go (if you don't do anything, it'll result in a large voltage spike), but my best guess is that in practice this is easier to deal with and redirect into a snubber cap, because of the small current drawn by each lamp (and therefore lower energy), and the fact that the inductance of the wiring is probably swamped out by each lamp's EMI filter inductance. I don't have any direct experience with trailing-edge dimmers.
"LED-compatible" does not mean "trailing-edge" in the USA: At one point, I bought a wide range of dimmers, from old to new, simple to fancy, to experiment with and provide a test platform for the company's lighting products. Not a single one of these used trailing-edge control, even the newest, fanciest ones that were advertised as being compatible with every LED lamp out there. Every single one used a triac, and a timing circuit made out of a combination of passives & sometimes some diacs.
What "LED-compatible" meant for the dimmer, is that (1) the timing circuit was relatively insensitive to load impedance, and (2) they used modern extra-sensitive triacs with low hold currents (a few mA, which was easy for the lamp's capacitance to sink @ 60 Hz).
Energy storage and dimmer compatibility in lamps: Good-quality non-dimmable LED lamps would use a bulk capacitor for energy storage, and then "peak-charge" it from the AC line, to give a fixed voltage directly across an LED string (sometimes with a simple current regulator). This bulk capacitor was for, as tooki said, storing energy across the 120 Hz cycle to provide a relatively steady light level.
Dimmable LED lamps would use a simple self-oscillating (2-transistor) boost or buck-boost converter to run open-loop at a constant(-ish) input current throughout the dimmer's on-time (therefore drawing well over the triac hold current the entire time) and dump that charge into an aluminum bulk capacitor connected to the LED string. Smaller conduction angle → less power delivered → lower voltage on bulk cap, less current through LEDs. The bulk cap is of course then responsible for filtering out 120 Hz ripple from the chopped input voltage, as well as the lower-frequency (1 Hz - 120 Hz) content created by the timing jitter in the dimmer, to prevent visible flicker.
The only other requirement for making a good dimmable LED lamp, besides the always-on converter and having enough bulk energy storage to smooth out the dimmer jitter, is having its EMI filter damped well enough (as mentioned above) against the leading-edge turn-on. In my experience, any LED lamp that does these 3 things will work well with pretty much any normal dimmer, whether listed as "LED-compatible" or not.
I have seen & taken apart plenty of bad-quality non-dimmable LED lamps though which save cost & complexity by using no bulk capacitor & no energy storage, just putting the rectified AC line voltage directly across an LED string (as tooki mentioned). They exist, they're out there, and they're AWFUL - the massive 120 Hz flicker is a bit too high-frequency to perceive directly, but any movement within their light creates all kinds of bizarre, distracting strobe-light effects. I hate these with a passion and any architect or landlord who buys these should have their license taken away.
Anyways, I've got a whole bunch of reverse-engineered dimmer schematics and simulations if anyone's curious.