Ensemble Darkhorse: Stewart Copeland. Rolling Stone readers voted him the fifth greatest drummer of all time in 2010.
Epic Riff: "Message In A Bottle" and "Every Breath You Take" are the most famous.
Face of the Band: Sting. The other two are still well-known and very respected, but his popularity eclipses theirs by a longshot.
Ho Yay: All three of them, when they weren't at each other's throats. Sting has remarked on the "sexual tension" between him and Copeland, and Andy Summers' biography details quite a few man-dates.
Misaimed Fandom: It's amazing how many people don't realize that "Every Breath You Take" is not a love song.
Never Live It Down: They fought a lot. In an interview before their reunion, Andy Summers lamented that this had become a cliche of the group.
Unfortunate Implications: The chorus of "On Any Other Day", in context, is just an extended whine about a really bad day, but it's easy to interpret the line "My fine young son has turned out gay" to mean "I'm disappointed that my son is gay". To be fair, there's nothing in the song to indicate the protagonist has a problem with his son's gayness - the key line "and it would be OK on any other day" makes clear that the sentiment is something more along the lines of "Son, you have the worst sense of timing ever," since the coda makes it clear the song takes place on the father's birthday.
Wangst: "King of Pain". It doesn't really have any overt whining or self-pity apart from the incredibly vague "but it's my destiny to be the king of pain", but the title is super angsty and the lyrics are full of little depressing vignette things.
The Woobie: The nameless main character in "Synchronicity II".