When is the last time tribal decks were competitive?
If you consider "Tribal decks" as decks with 


  core cards in the deck [that] care about a certain creature type, 


there's plenty of examples out there.

This will be a very broad answer that'll brush over many specifics of the metagame and the differences between physical and digital plays, focusing only on information available from resources such as MTGTop8, which accounts mostly for MTGO tournaments.

In the current (as of 09/09/19) standard meta, Orzhov Vampires represent 11% of the (MTGO) field, while Jund Dinosaurs represent about ~5%.

Both of those decks focus on aggressively dishing out their creatures and hitting for as much damage as possible, using those same creatures to create card advantage (such as Champion of the Dusk and Ripjaw Raptor), or decrease the clock to end the match more swiftly (such as Legion Lieutenant, Sanctum Seeker, Marauding Raptor and Rampaging Ferocidon).

Both these decks are currently posting good results consistently in standard play, as you can see from the links provided.



When talking about Modern, the obvious examples are 5 Color Humans which focus on Human creatures, while some Eldrazi variants focus on Eldrazi creatures and fast mana. It's also worth mentioning the Spirits deck, which focuses on Spirits. All three of these currently post good results in the modern metagame, although they're not as dominant as they once were, especially the eldrazi decks.



Legacy has always had some tribal decks being represented, although not many with expressive results. The currently most represented tribal decks are Goblins, Humans and Slivers, although they're only about ~4% of the whole Legacy field, which favours more controlling and combo decks due to faster mana and cheaper countermagic.

As @Veskah noted, Elves has been very represented in Legacy at some point, but its power has dwindled lately due to the more controlling meta for the format. That said, Elves is one of the original tribal decks in Legacy, and it veers from the usual strategy by being something of a combo deck, relying on generating huge amounts of mana and putting many creatures on the board to finish the game with a Progenitus or a Craterhoof Behemoth.



Vintage has also has some tribal decks in the meta for a while, specifically Fish, which used to be Merfolk tribal decks, although today the term has been adapted to mean any kind of (mainly blue) deck with cheap creatures and lots of countermagic. Along with Fish comes the ever-present Eldrazi, although once again it's more of a hate/control deck than an actual tribal deck.