i recognized that this is a very old post, however it fits exactly my problem - i can't delete my playlists within the desktop client of spotify. i already deinstalled it completely, downloaded the latest version and reinstalled it again - same problem.
in right-click context menu is no entry about it (see screenshot).
(press DEL-trick also don't work, cause playlists never marked as 'selected')
Furthermore, most of my Playlists just don't load their names for about a week now (not even after reinstall)