There’s the ABUR dual lands, then the Shocklands. But after that what are the game’s third best dual lands? Are the checklands widely regarded as the game’s third best dual lands or is it the painlands? Or is it something else?
depends, I'd say tango lands, then bicycle lands for control decks (I'm assuming these are being played with fetches), fastlands or painlands for aggro. If fetches aren't being used, I'd go with fast lands and painlands/checklands.
Eh, fetches in vacuum are mediocre for fixing. They are great if you just need to make use of the shuffle or stock your graveyard, but the fact that they only fix to one color at a time is a major drawback for decks that have a lot of CC or 1CC cards. Their power comes from being paired with ABUR duals or shocks. Also, a call out to bouncelands as the best fixing in pauper.
If you had to choose either fetches or ABUR duals, would fetches still be your #1 pick?
Most Vintage and Legacy decks run more fetches than ABUR duals, and abuse the flexibility to get basics instead of ABUR duals (to dodge nonbasic land hate). Modern and Budget Legacy decks have roughly as stable manabases without any ABUR duals, just using fetches with shocklands instead. Even Standard used to have very stable multicolor manabases using fetches with tangos (Khans + BFZ block).
A common counterargument is that fetches are only good if you also have ABUR duals or shocks, otherwise they're bad. But even many monocolor decks run them (e.g. Elves, Burn)! They're also useful for:
Depends on your deck. Wizards intentionally printed these lands differently so they all had pros and cons, instead of making a hierarchy where one was clearly better than all others. That opens up design space.