Impossible to install single-user version of RVM

Indeed, I solved this by uninstalling old versions of RVM: sudo rvm implode and then deleting the file /etc/rvmrc. Once done, I could install the single-user version and everything worked fine!

Sorry Remear, I wanted to edit your answer or complete it via my comment but I could not (comment can only be edited within 5 minutes...at least I upvoted...).

Once you have a system-wide install, you can't run a single-user install as it will detect your system-wide install first and try to update it.

But it is possible to have a root install & then user install by specifying the installation path :

curl -sSL https://get.rvm.io | bash -s stable --path $HOME/.rvm

You will also have to set the single-user install path in your user path manually as RVM does not create it when you already have a system-wide install (not really an intended use) :

# .bashrc
PATH=$PATH:$HOME/.rvm/bin # Add RVM to PATH for scripting

And also load your user rvm :

# .bash_profile
[[ -s "$HOME/.rvm/scripts/rvm" ]] && source "$HOME/.rvm/scripts/rvm" # Load RVM into a shell session *as a function*

Are you setting rvm_path in ~/.rvmrc, or in /etc/rvmrc, or in one of your bash scripts? I'd recommend removing both of those files as well as ~/.rvm and then try installing rvm again WITHOUT sudo.