I've just got Arch Linux installed on a USB drive. I set it up so it works with both UEFI and legacy BIOS boot. It works great - I can install whatever tools I need on it, I can even use it as a little portable Linux drive if I need that. I've also used it to install Arch on a number of my computers.

The only issue is that the USB drive is pretty slow, so it's not the best experience using it

I've been using SystemRescue (formerly SystemRescueCD) since forever & it's usually more than enough for anything I need to get done. Though nowadays just about any Linux live image can probably do the job if that's all you have at the moment.

I mostly use systemrescueCD, but in fact it doesn't really matter. All I usually need is to mount the root filesystem, dev, sys and proc, then chroot into this environment and fix the breakage. Any live image will work for that.

I just use whatever live image I last wrote to a flash drive, because it requires zero extra effort or storage, and they all have what I would need from a rescue environment anyway.

I use a Puppy Linux derivative - QuickPup64 to be exact. What sets this apart from other Puppies is that QuickPup is updated regularly and has a recent kernel, with the biggest advantage being that QuickPup is compatible with almost all the popular Linux package formats out there.

Wider package support. Aside from PET, Debian (DEB), Red Hat (RPM), Slackware (TXZ, TGZ), and Arch Linux packages (PKG.TXZ).
It also accepts the following packages (natively installed, no more package conversion needed. This allows to utilize almost all available linux packages):
->Slitaz (*.tazpkg)
->Tiny Core (*.tcz,*.tce,*.tcel,*.tcem)
->Pardus (*.pisi)
->Alpine Linux (*.apk)
->Slax (*.sb)
->OpenWRT (*.ipk)
->0linux (*.spack)
->Void linux (*.xbps)
->Damn Small Linux (*.dsl)
->Puppyrus (*.pfs)
->Nutyx (*.cards.tar.xz)
->Solus (*.eopkg)
->Stampede (*.slp)
->Porteus (*.xzm,*.lzm)
->Gentoo precompiled package (*.xpak)
->Paldo Linux
->Venom Linux (*.spkg.txz)

I also have to say that Puppy is fast - it's designed to be run off USB and work on older systems with limited resources, so this thing absolutely flies, compared to a regular distro.

Gparted-live or base Archlinux for me as I have the most experience with these 2

I have a drive with Ventoy on it and ISOs for Hiren's, Rescatux, and NixOS, along with a custom ISO of NixOS with some tools pre-installed. Super easy to generate.

I have a Porteus USB that I keep handy with my old Hirens USB. It’s Slackware based and a bit goofy when it comes to installing other apps, but because it’s a USB portable OS first I feel it’s one of the ones that handles that limitation best. I used to use Puppy IIRC.

I'm using Hiren's Boot CD (not PE), it has a handy linux rescue environment and password reset utility.

But if it doesn't count, recent I found MX Linux installation USB are also useful for me to do partition tasks, as well as grabbing files from folders with limited privileges of my Windows partition.