The most popular lies from the terminally online that I have noticed:
Proponents of censorship claiming to support free expression
Authoritarians claiming to believe in human rights
Warmongers claiming to be pacifists
Science-deniers claim to love science
The worst is dealing with people who claim to invite honest discussion, but then have nothing to say when a perfectly polite detractor asks pointed questions, and instead respond only to litanies of insults from shitlords like me... almost as if these people care more about their image than anything else.