In the UK I think

"between a rock and a hard place"

"between the devil and the deep blue sea"

"between the hammer and the anvil"

are the most common they would all be understood as having only two choices never of them good.

A similar one now current in the UK is "damned if I do and damned if I don't"

can be applied to second or third person.

"damed if he/she does and damned if he/she doesn't"

"damed if they do and damned if they don't"

Currently being said about our prime minister on various news media outlets and social media who has not gone to visit flood victims.

he is going to get "stick" = a common idiom for criticism whether he now does or doesn't go to visit.

stick for not going immediately stick for going now it is late in the day. [ we say late in the day it does not have to be the same day always]

I can only speak for myself (Canadian EN), but I wouldn't use any of those alternatives, and I don't think I've ever heard anybody actually use them. Although "between the devil and the deep blue sea" rings a bell, it sounds rather tongue in cheek to me.

I'd even have a hard time using "caught between a rock and a hard place" without it feeling rather cliché.

Maybe it doesn't have quite the same meaning, but there's also "damned if you do, damned if you don't". You could shorten it to "damned if you do".

You could also try to squeeze in the phrase "a no-win situation" somewhere.

I was just using the polite form of the phrase.

Another way of expressing a hopeless situation would be "up the creek without a paddle."

Also, "going the wrong way on a one-way street."

Caught between a rock and a hard place is common among my age group. I still use it today. I never heard that story about the miners. To me, the phrase means you're caught between two things that cannot be moved. In other words, you're trapped, and without any good options.

I heard "caught between the devil and the deep blue sea" many times on ships, and I used it in that environment as well. Sailors claim it as a nautical terms (but who knows, sailors claim just about everything as a nautical term). The "devil" is a long difficult seam on a wooden ship. If you're caught between the "devil" and the deep blue sea you are usually hanging over the side in a precarious position. Incidentally, the phrase "I've the devil to pay" is also a nautical term. See what I mean about sailors! When you caulk a seam on a wooden ship you are "paying the seam". That is the process of sticking oakum in the gap and covering it with tar. The full phrase (to a sailor) is "I've the devil to pay and no pitch hot", which means "I have a lot of work to do but nothing is ready!".