Bishop, Knight & King versus King.

Can you read?

http://chessskill.blogspot.com/2017/05/bishop-and-knight-checkmate.html

Yes. At home for this endgame, or, for harder ones http://tablebase.sesse.net/ has Nalimov tables you can practice against for FIDE basic rules mates or positions that don't need an altered mate to circumvent the 50 move rule. For those that do https://syzygy-tables.info/ uses Syzygy tables (but good luck with those). Only up to 7 men in total at the mo though.

You can use Wilhelm to generate mate positions of various depths, available at http://web.archive.org/web/20081208015339/http://www.geocities.com/rba_schach2000/overview_english.htm.

Look through the 252 articles here and tell me whether anything is missing. http://chessskill.blogspot.com/search/label/checkmate%20patterns

There are about a couple of thousand endgame classifications up to 7 men, so probably yes.

Yes. Here on chess.com.

And trillions or billions of trillions possible positions.

You can even 'work out' with various piece setups against the chess.com computers.
You choose the setup.
But I don't have the link handy. I believe I posted one earlier and it was here. Years ago.

By the way - you can't checkmate with K and bishop against King if there's nothing else on the board for either side.
You can't even set up a checkmate with just that.
Same with K plus knight against King and no other material for either side.
Insufficient material.

If one adds a pawn to the lone King in those bishop or knight situations - do mates by the bishop or knight become set up -able?
Some do. With 'rookpawns' ...
But maybe the computers still assign 'insufficient material' there.
They do with K plus knight versus K plus knight even though mate can be set up. It can be blundered into also
.
Same with any King plus minor versus King plus minor.
Mate possible but its still insufficient unless you've done it.
Is a pun in order? Illegal for minors under 14 to 'work'.

Its extremely extremely rare but there are some positions where King plus knight versus King plus rookpawn in which the King plus N can force mate.
If the flag falls on the side with K plus rookpawn does the side with the Knight get the win?
Maybe only at Night ...

Even with 2 pawns the knight can force a win. Unfortunately with this site's mating material vs timeout rules, black could deliberately let his time run out after nf1 and obtain a draw, because white only has a knight. This is a good example of why the rule should be "checkmate has to be impossible" for a draw to be declared, not simply the piece count. Another good example of this is this endgame:

The site will declare a draw before white can play Nc7

@EndgameEnthusiast2357
Yes - there's these horrible situations where the losing side can let their own clock run out to get a draw instead of a win.
And whether its computers on a chess site or tournament directors applying inadequate rules or just coffeehouse and park players arguing about whether its a win or not and who has to get up from their seat to let in the 'I'm Next' player ... its an ugly fact of life in the chess world it seems.
When masters train prodigy pre-teen players perhaps they tip them off to such unpleasant realities. Arming them with pre-knowledge.

That's why the FIDE rule that if checkmate is possible in any way, it's a loss for the flagging side, that's most logical. Here are 2 more absurd examples of how losing on time can improve your game result (for black).

Same here. The player wasn't able to mate his opponent inside 50 moves. I think he knew it was possible but didn't know the technique required for this. I remember asking a group of players if they could do, admitting freely that I certainly wasn't sure if I could remember it, but adding that I could at least do two bishops and a king against king, at which point some a$$hole challenged me to do it publicly in front of several other players against him (me with the king and bishops, him with the king). To be honest I hadn't done it in a while and felt a bit nervous, but once I started I mated him easily enough. Another player then said he should show me that he knew how to mate me with king, knight and bishop against my king. Unfortunately while he was trying to do this, he had some help from a kibbitzer who knew the way to do it which annoyed me quite a bit, given that he had been trying to show me up, and therefore should have been made to put his money where his mouth was.