First off, we use a trip line less than 1% of the time and only in places that we think may have junk on the bottom, such as commercial harbours. And in over 40 years of anchoring we have never had an anchor saved by a trip line and we have never lost an anchor.
We have got an anchor fouled, probably half a dozen times but have managed, on every occasion but one, to get clear by shortening up the rode, engaging the chain break to unload the windlass, and then pulling with the engine from various directions until we pulled the anchor clear. (You need strong gear for this game as the loads can be prodigious.)
On the one occasion that did not work, we got clear by lifting whatever we were snagged on a bit off the bottom, using our massively powerful Ideal Windlass, and then letting the chain run, which popped the anchor out from under whatever it was.
One other point. Your risk of getting your anchor fouled on an obstruction goes way down if you use one of the new design anchors like a SPADE or a Rocna in comparison to an older design like a CQR. The reason is that the former set in their own length, but the latter usually drag for at least 20-feet before setting, thereby increasing the chances of finding something to hang up on.
For the occasional time that we do use a trip line, this is how we do it: