I'm using PuTTY because I prefer native software. That said, it seems that Microsoft started shipping its own build of OpenSSH with Windows 10, so probably that's not needed anymore.
Another approach is to enable "Use Ctrl+Shift+C/V as Copy/Paste" in terminal properties and just use those shortcuts, they seem to work fine in ssh.
But I find PuTTY's terminals to be almost worse. The visuals are seriously eye cancer inducing, and PuTTY's tools don't seem to be built with a terminal-based workflow in mind.
So, I much rather use either MS' OpenSSH port (on Win10) or MSYS 2's (which is included in the Git Bash for Windows, too).