Change Xubuntu's default text editor globally

If you do not want to install additional applications you will have to manually edit ~/.local/share/applications/mimeapps.list

Under the [Added Associations] section add:

application/x-perl=sublime.desktop
text/plain=sublime.desktop
text/x-chdr=sublime.desktop
text/x-csrc=sublime.desktop
text/x-dtd=sublime.desktop
text/x-java=sublime.desktop
text/mathml=sublime.desktop
text/x-python=sublime.desktop
text/x-sql=sublime.desktop
text/x-tex=sublime.desktop

Just to follow up on bodhi.zazen's answer with a more complete file:

[Added Associations]
application/ecmascript=sublime_text.desktop;
application/javascript=sublime_text.desktop;
application/mathematica=sublime_text.desktop;
application/x-chess-pgn=sublime_text.desktop;
application/x-ccmx=sublime_text.desktop;
application/x-ica=sublime_text.desktop;
application/mathml+xml=sublime_text.desktop;
application/sql=sublime_text.desktop;
application/x-perl=sublime_text.desktop;
application/xml-dtd=sublime_text.desktop;
message/partial=sublime_text.desktop;
message/rfc822=sublime_text.desktop;
message/news=sublime_text.desktop;
model/vrml=sublime_text.desktop;
text/x-java=sublime_text.desktop;
text/xmcd=sublime_text.desktop;
text/x-vhdl=sublime_text.desktop;
text/x-subviewer=sublime_text.desktop;
text/x-adasrc=sublime_text.desktop;
text/x-modelica=sublime_text.desktop;
text/x-objcsrc=sublime_text.desktop;
text/x-ssa=sublime_text.desktop;
text/x-mpsub=sublime_text.desktop;
text/x-txt2tags=sublime_text.desktop;
text/x-svhdr=sublime_text.desktop;
text/x-verilog=sublime_text.desktop;
text/x-troff-ms=sublime_text.desktop;
text/x-mof=sublime_text.desktop;
text/x-readme=sublime_text.desktop;
text/x-uil=sublime_text.desktop;
text/x-scheme=sublime_text.desktop;
text/x-install=sublime_text.desktop;
text/x-log=sublime_text.desktop;
text/x-ooc=sublime_text.desktop;
text/x-uuencode=sublime_text.desktop;
text/x-ms-regedit=sublime_text.desktop;
text/x-ldif=sublime_text.desktop;
text/x-mup=sublime_text.desktop;
text/x-makefile=sublime_text.desktop;
text/x-nfo=sublime_text.desktop;
text/x-lua=sublime_text.desktop;
text/x-pascal=sublime_text.desktop;
text/x-cmake=sublime_text.desktop;
text/x-dsrc=sublime_text.desktop;
text/x-rpm-spec=sublime_text.desktop;
text/x-tex=sublime_text.desktop;
text/x-markdown=sublime_text.desktop;
text/x-microdvd=sublime_text.desktop;
text/x-idl=sublime_text.desktop;
text/x-scala=sublime_text.desktop;
text/x-setext=sublime_text.desktop;
text/x-python=sublime_text.desktop;
text/x-literate-haskell=sublime_text.desktop;
text/x-copying=sublime_text.desktop;
text/x-ocl=sublime_text.desktop;
text/x-dsl=sublime_text.desktop;
text/x-texinfo=sublime_text.desktop;
text/x-dcl=sublime_text.desktop;
text/x-fortran=sublime_text.desktop;
text/x-gettext-translation=sublime_text.desktop;
text/x-erlang=sublime_text.desktop;
text/x-gettext-translation-template=sublime_text.desktop;
text/x-emacs-lisp=sublime_text.desktop;
text/x-iptables=sublime_text.desktop;
text/x-haskell=sublime_text.desktop;
text/x-lilypond=sublime_text.desktop;
text/x-patch=sublime_text.desktop;
text/x-matlab=sublime_text.desktop;
text/x-reject=sublime_text.desktop;
text/x-tcl=sublime_text.desktop;
text/x-svsrc=sublime_text.desktop;
text/x-troff-mm=sublime_text.desktop;
text/x-moc=sublime_text.desktop;
text/x-troff-me=sublime_text.desktop;
text/cache-manifest=sublime_text.desktop;
text/x-eiffel=sublime_text.desktop;
text/enriched=sublime_text.desktop;
text/htmlh=sublime_text.desktop;
text/rfc822-headers=sublime_text.desktop;
text/plain=sublime_text.desktop;
text/troff=sublime_text.desktop;
text/x-vala=sublime_text.desktop;
text/css=sublime_text.desktop;
text/x-c++hdr=sublime_text.desktop;
text/x-c++src=sublime_text.desktop;
text/x-chdr=sublime_text.desktop;
text/x-csharp=sublime_text.desktop;
text/x-csrc=sublime_text.desktop;

What actually helped was to uninstall Mousepad. I was not using it anyway. All the files are now opened in Sublime.