Are there named opening variations that include not castling?

I look for no castle opportunities in my games and was curious if there are standard openings.

Well, there is the Berlin Defence with queen trade (along the lines of 1. e4 e5 2. Nf3 Nc6 3. Bb5 Nf6 4. O-O Nxe4 5. d4 Nd6 6. Bxc6 dxc6 7. dxe5 Nf5 8. Qxd8+ Kxd8). Not sure if that's the sort of thing you're looking for, though. And there are some lines in the Sicilian and French, but I forget what they're called exactly.

But in most games it's a good idea to castle at some point, unless you have something like a stonewall with locked pawns in the centre... then it can be better not to.

Here is one: neither side castles

Here is another:

PantedakelisCounter sac is sacing Ur castelings after e4e5f4f5exf5 De7 Dh5+ Kd8

But also Jobava London with f3 g4

It's a strange question because most openings are defined after only a few moves. For example 1 c4 English or 1 e4 e6 French. Neither side has castled and there is no obligation to do so but it's usually advisable.
This is true even of opening subvariations, for example 1 e4 c5 2 Nf3 d6 3 d4 cxd4 4 Nxd4 Nf6 5 Nc3 a6 6 Bg5 e6 7 f4 Qb6 8 Qd2 Qxb2
This is the Sicilian Defence, Najdorf variation, 6 Bg5, Poisoned Pawn subvariation. You may or may not castle but the opening name does not change.

Here is another example

Elshad's oppening

@TheOnlyBenJo said ^

PantedakelisCounter sac is sacing Ur castelings after e4e5f4f5exf5 De7 Dh5+ Kd8

I think he meant sound openings.

Here is another example