Wandering Minstrel is almost always the most useful village card for engines in the currently released expansions. By providing 3 cards worth of filtering and information for the next draw or hand, it greatly improves the consistency and effectiveness of engines, particularly those that have slightly limited draw options (+2 Card draw, for instance) but any other form of draw as well. Cantrips benefit greatly too, as knowing what you're getting can be the difference between continuing a chain and hitting a dead card. Wandering Minstrel can be riskier if you rely on drawing Victory or Treasure cards for some purpose, but even in decks that need those things, the Minstrel can assist in drawing the entire deck, which means skipping stop cards for later was probably better.
One of the drawbacks of Minstrel is that since it cycles 4 cards total, it can trigger unwanted reshuffles. As with any other card that carries this risk, use your judgment. If your current hand has too many good cards you don't want to miss the reshuffle, you should think about holding off on pulling the trigger. On the other hand, if your hand sucks you can always go ahead and trigger that sucker with a grin on your face.
Sometimes people open Minstrel. Like any village, it's often pointless since you don't have enough actions to play anyway. However, if you need to play a key card regularly, skipping past 3 coppers/estates and getting to the next shuffle quickly can be a good option. I would basically never open Silver/Wandering Minstrel, but other options such as Steward/Minstrel are completely reasonable. Steward is a card you want to play a whole lot early on, whether it's for trashing or hitting $5/6, and anything that helps you get there is a plus.
You should skip Minstrel if your deck is almost entirely treasure (e.g. in a Big Money deck), but that goes without saying. It's sort of useful to filter through junk cards, but the risk of skipping your Gold tends to be too much to make Minstrel a good option. This all goes without saying, but I bring it up because some players buy Minstrel for the cycling without having an end plan, and if your end plan relies on drawing treasures you don't have the draw far, you are probably making a mistake getting a card that efficiently discards those.
Minstrel is also very good for cards that improve in power with knowledge of the top of the deck, such as Mystic and Wishing Well. It's a cheap enabler for them, and you were probably buying it anyway.
While decks that cannot trash like the filtering from Minstrel, make no mistake: decks that trash want it to. They benefit just as much from knowing what cards are coming next, especially if you need to make a decision with a terminal draw card or cantrip. Wandering Minstrel is a swiss army knife that you should be considering buying almost every time it's in the Kingdom.
Some specific interactions to look out for: (I tried not to be exhaustive but this card is so good I thought of a lot)
- Draw to X cards. Sometimes your final draw amount is limited in a deck that relies on these, so it's useful to get past the crap and know you have a couple of useful things coming up.
- Ruins. Ruins counter Minstrel if you don't have a method of trashing them and can't use them for anything, so be careful. Minstrel is still very good, so you should probably compete in the Ruins split instead of avoiding Minstrel.
- Baron. Be very careful with this combination, as it's pretty likely you will skip the Estate you need when spamming Minstrels. If you have another form of draw, it's good to make a reasonable guess where the Estate will be and try to draw it with a card other than Minstrel.
- Minion, Outpost. If your next 'hand' is going to be 4 cards, you'd love to have a way of ensuring at least one of them is good. Minstrel offers you that insurance or perhaps suggests you should just take the $2 from Minion now and save things for the next turn.
- Torturer, Cultist, Treasure Map, Herald. Chaining these is rarely easier than it is with Wandering Minstrel.
- Ambassador. Every now and then it's worth opening WM/Ambassador or at least getting a WM shortly after the game starts. In Ambassador games, you tend to be overrun with junk cards early, and WM can get you to playing the all-important diplomat sooner and more often.
- In Ghost Ship games, WM makes a decent soft counter since you can discard a topdecked Estate or something. However, it's best to have some powerful draw to play immediately afterward so you're being proactive.
- Potion. If you need to buy cards that use Potion, you might hold off on the Minstrel until later. Odds are, if you made the investment of buying the Potion, you had a good enough reason that you need to be careful about this.
- Scrying Pool, Golem. Minstrel helps you set up big Pool draws or at least clears more junk from the top than Pool ever will. One of the best villages for a Pool deck. It's also wonderful for getting some guaranteed hits with Golem.
- Philosopher's Stone. Just don't do it. Skipping a Stone with your own Minstrel is miserable, and pretty likely to happen. This applies to any treasure card, but normally PStone benefits from cycling cards, so this is a very notable exception.
- Possession. Be wary! Cycling 4 cards means a possessor can quite easily trigger a reshuffle on you, and if you have lots of Minstrels you are likely to have lots of bad cards in your newly shuffled deck. You can still play Minstrel against Possession, but trashing is almost a must and a crafty opponent can Possession lock you if your deck is smooth enough.
- Counting House. This can be a powerful combination. Discard your Copper while cycling, get to the bottom of the deck, (using Minstrel to guarantee CH is in hand) redraw the Copper.
- Peddler. Minstrel is great for lining up actions to buy Peddler, as well as to help smooth out the Peddler-driven economy later on.
- Tunnel. Minstrel makes getting Gold from Tunnels easy, but it also makes you skip Gold a lot. I suggest only playing this deck with support from another draw card.
- Hermit. A light interaction, but Minstrel does often discard cards you can trash with Hermit directly after.
- Rebuild. Minstrel is great in Rebuild decks as long as it isn't triggering terrible reshuffles. It gets you to Rebuild faster, skipping Copper and telling you a few of the victory cards that you are now past, and your goal is generally to just play Rebuild with some idea what you'll hit, so that's a good thing.
- Advisor, Envoy. Give your opponent nothing but good draw options, why don't you.
- Prince. Wandering Minstrel tends to be a fantastic target for Prince. Usually a Village isn't as good as something like a Smithy, but Minstrel does so much for an engine that opening a turn with it is a great choice. As usual, you must first determine if the Prince was worth it, though, and there may well be better targets in your hand.
My personal rating for Wandering Minstrel is 8.5/10. It's one of the most versatile, most useful cards in Dominion, but there are some situations where it's plain bad and, being a village, it doesn't 'accomplish' as much as cards that provide draw, buy, and coin, so you need other cards to combine it with. Fortunately, there are a whole lot of those. At $4, it's always an easily affordable option, so it's usually open season on this one. Bard power!!!