I honestly don't watch much modern anime, just for this reason. I'm all for using tropes when it works, but the overuse of tropes in anime is just ridiculous.
With 80/90s anime there were plenty that sucked and were full of cliches, but there were just as many that were trying to do their own thing. I don't mean to sound like an old fart who goes "BACK IN MY DAYYY-" but I seriously have tried to find new anime I like, and I just have come across VERY few that stand out. I don't find them as diverse anymore, in style or content.
A lot of people here named cliches I hate already, but here's a few others that bug me....
1: The 'special' protagonist who always finds his way out of things through use of unknown hidden powers, power upgrades, or just being so darn special that he destroys his enemies with sheer awesomeness.
Sooo many animes with this now....I did watch some of Attack On Titan at one point, which seemed promising, but the moment I found out the main character transformed into a Special Titan, I lost the urge to check out more. It just became any given anime to me at that point.
2: School settings....and pretty much all the tropes that go with it.
Aaauuughhh....I grew up with Ranma1/2, Sailor Moon and Project A-ko and I still can't take anymore of this.
3: Reused Animation for Transformations / Special Attacks
Oh my god, I didn't even like it in Sailor Moon, and now they don't even just use this for magical girls.
4: Abuse of silly, over the top anime expressions.
No, not all anime has to abuse the silly expressions just because it's anime.
5: Eyesore designs.
When a character has an insane hairstyle or is just so loaded with out of place details that I go, "What am I even looking at right now?!"