By far, the biggest use for this is hitting $7 when you want to buy a $5 more than golds or any other option that might be in the Kingdom. If you are buying a Witch, well get her here 40% of a turn faster (just do not trigger a bad shuffle) & setup a turn to maybe buy a Rc or something else. Likewise, if you are Duchy dancing, then toss some green so you can actually hit $8 next turn.
You pretty much always want this when you have more cash & no viable alternative unless you are triggering bad reshuffles with it or maybe if you expect to be hit by Minion without being able to cycle back to good cards if you discarded them.
Beyond that, it is handy to setup Engines if you can get your kickoff components (e.g. Pool, Village/Smithy, etc.) into an empty discard/draw pile with just enough cards to force a 5 card shuffle with the event. It can be particularly nice if you can trigger shuffles while holding lousy cards like CCEEE.
So now for the fun stuff, where do you actually want this instead of other stuff that can help:
1. Rebuild. You quickly reach a point where more Es are dangerous, Silvers are not too likely to actually be played, and there is decent odds that you will have your Rebuild targets in hand when you have a Rebuild or two you want to play. By midgame, spare $2 and $4s should be used to setup strong hands to minimize flipping your rebuilds and also to eliminate hands like RebuildX3/EstateX2
2. Off a Kc/Scav deck to force a Shuffle so your Kc/Scav pairs are never in the same hand (only happens when you hit $2 in the two cards you do not control, but best 2 of 3 helps with that as well).
3. Deck control for Chouse decks. Dump the coppers, setup Hamlet/ChouseX2; churn to the end of your draw deck if you have 1 buy and no more Chouses left before the shuffle (e.g. Chouse for $12, churn 7 cards to get back to Province buying a turn sooner).
4. Removing green/junk from deck top when needed - Countering Rabble & friends, moving away unwanted Prizes, keeping only the important treasures (like Quarry) when you have bought a Mandarin.
5. Being really mean after your last Possession turn (double Pillage for $2, joy; pretty much nothing but dead draw for $4).
6. Fishing for high variance cards (e.g. Plats, Pstones, banes, Curses against a single Mountebank, Library/Hground in BM games).
7. Knight/Giant/Sab/Swindler defense when you will subsequently stack the top of your deck 5 deep (e.g. Alchemist, Treasury, Wt)
8. Traveller/Distant lands/etc. accelerator.
9. Post Tmap to fish for useful cards (e.g. +buy) or to cycle past useless gold.
10. Increase alignment odds, particularly with deck tracking (e.g. Tournament, Mint/Plat, Province/Explorer, Menage/Uniques).