Beam Me Up, Scotty!: As listed under Memetic Mutation on the YMMV page, one of Paul Lynde's most-quoted zingers, usually done by the quoter with Lynde's trademark mannerisms and voice, is "Because chiffon wrinkles!"note given to the question "Why do Hell's Angels wear leather?", despite that not being what Lynde actually said — what he actually said was "Because chiffon wrinkles too easily."
H2 for the last two seasons of the Bergeron edition, due to that being its new logo ("Hollywood Squared," get it?).
Giant Bomb: [during a review of the 2010 game] So we're actually playing Hollywood Hollywood.
Keep Circulating the Tapes: The Peter Marshall-hosted versions were believed to have been largely destroyed until around 2002, when several hundred episodes (the number varies- Peter Marshall said in a webchat there were 3500 episodes they stumbled across) from both the daytime and nighttime versions were found unintentionally as part of a search for episodes of Dark Shadows, of all shows (read more here). Even then, GSN aired them only from 2002 to 2003.
The Davidson version was rerun on the USA Network for a few years in the '90s and hasn't been seen since.
Some episodes of the Bergeron version weren't rerun on GSN due to celebrity clearance issues, including the 1998 Comic Relief week (though at least one ep of that circulates, taped from the satellite feed).
Long Runner: The original NBC version ran for 14 seasons and the 1970s syndicated version ran for a decade.
No Budget: Hip Hop Squares, which offered a top prize of $2,500 in 2012. Could be justified, as it was produced for MTV2, a cable network (plus most of the budget likely went towards getting the celebrities).
Screwed by the Network: The NBC version. Despite being the number one game show in daytime at the time, then NBC president and CEO Fred Silverman abruptly cancelled it, after moving its time slot around several times, along with Chain Reaction and High Rollers, to make room for a 90-minute talk show hosted by David Letterman. Reportedly, Silverman had passed on the show back when he was vice president of daytime programming at CBS, and after seeing it become a big hit on NBC, was obviously not happy.
Troubled Production: The Vegas episodes. Paul Lynde was frustrated by the supposedly inferior lodging facilities in Vegas (among other things) and in general soured the overall mood. Not only that, there wasn't much time to play the game, some contestants were tourists (and drunk, to boot), and some celebrities, particularly George Gobels, would often rush off to gamble.
According to Peter Marshall, Bert Parks was originally chosen to host the NBC version. Parks hosted the original pilot, but Marshall subbed for Parks on a second pilot and wound up being chosen when the series was picked up.
Marc Summers was slated to host the 1998 revival but was dismissed for fears that his OCD (which he had revealed on The Oprah Winfrey Show a year prior) would make it difficult to work with him.
Prior to Whoopi coming on board, Roseanne was attempting to create the Planet Hollywood Squaresin the mid 90s;this Game Show Forum thread revealed that it was intended to be paired with a revival of The Gong Show (which did in fact have a pilot taped), but held off until 1997 to see if their revivals of The Dating Game and The Newlywed Game were working out; by that time, Roseanne presumably had pulled out and PH was in deep trouble (having overextended themselves with far too many locations and crazy concepts).
You Look Familiar: John Davidson was a semi-regular panelist on the Marshall Squares before becoming host of the third version.