AMVs, and some other fanvids, that I've liked. Some I think are genuinely good, or at least strongly appeal to me; others are listed out of collector-instinct, especially if they're technically impressive. I've tried to sort them, 'good' first in a category, followed by the 'collector' ones I didn't like as much but didn't want to lose.
Link update 25 Dec 2020.
Not all links will work, as videos get deleted from Youtube, but I keep the text in hopes of finding them again. Sometimes it works!

I save the good stuff, so 'broken' stuff can be made available if you know me. Broad appeal (all good)

Right Now (on people discovering anime)

Anime Munters Foe Yay -- antagonist implied yaoi humor.

Ship Happens. Funny, and pretty painstakingly done if you watch carefully.

Chronicles of Ghibli -- Narnia/Ghibli across, amazing cut work.

Soldier A. Original tribute to redshirts.

Lucky Me (Buffy romance) (not anime)

Golden Boy/I Want to Ride My Bicycle -- first AMV I liked; I liked it without knowing the sources, either the anime or Queen or the cover band.

Princess Tutu Hold Me Now -- demonstrably works for people who haven't seen Tutu. Also, spoilery.

As above, with "disco" visual effects

Watchmanga (Watchmen, Darker Than Black, Cowboy Bebop)

Caramell Dansen misheard lyrics

Silence of the Vampires -- Hellsing cross Silence of the Lambs. I haven't seen either and liked it. OTOH I did know something about Hellsing.

Space Girl (not anime)

Fireflies, random cheerful melange

Complete History Of The Soviet Union, Arranged To The Melody Of Tetris Not anime, but classic.

Touhou Bad Apple (silhouette)

Touhou Bad Apple (English subtitles in closed caption) (silhouette)

Ballad of Mike Haggar Video game fanfic.

The Phantom Menance: How to fight with lightsabers

Is YOUR daughter a magical girl?

Chobits / The Internet is For Porn

We Need You various things to Starship Troopers. Should be good for all. Also (higher resolution?)

I liked (so all good) but think you might need more specific background, but they don't need their own fandom categories

FMA "Date R*pe" -- "funny"

Utena Matrix (broken even there?)

One-eyed fish, Mushi-shi AMV

TTGL x Mulan Also this (broken) is similar.

Sora no Woto Good Day to Die

Moe Division

Captain Harlock live action fan trailer

Cleaver Girl Higurashi's Rena to "Creepy Stalker Girl". Excellent, might be decent if you haven't seen it.

Lodoss War Saga begins -- the Weird Al Star Wars song set to Lodoss visuals.

Evil Buffy opening (not anime)

Princess Tutu Caramelldansen -- with real dancing!

Princess Tutu "Extra Episode" (broken)

Macross Frontier Starship Troopers -- has no soul but very well executed.

Bleach 300 Trailer. There's a zillion of these, but this one features Kenpachi's 11th Squad as the Spartans, so it has a thematic unity most others lack. At least if you've seen enough Soul Society arc to get a feel for Kenpachi and his squad.

Hermione the Pirate Queen

Scrapped Princess: Me Against the World. 10/10 for theme. :)

Senshi on Springer

Wizard of Ozaka

Them Daleks Ain't So Scary

Torchwood Banned From Argo. Yeah the Torchwood crew would get themselves banned.

This Land Is Mine war video

Blank Space -- Death Note vid cut to go with a song in parallel. Fiddly now, you'll need the audio in a Youtube tab.

Yona One Match Fight Song

Haruhi Suzumiya

Harassment of Kyon -- Koizumi/Kyon/"I want to be your girlfriend"

Haruhi/She's Just Oblivious -- (high quality version) very well-matched if you know the series.