"Be My Girl - Sally" is at first a happy pop tune, then turns into a bizarre poem read by Andy Summers about a blow-up doll, then turns into a painful shriek trailing off into the distance...
"Masoko Tanga", which is the Last Note Nightmare for their first album. Sting has (perhaps jokingly) referred to being hypnotised while playing the song, saying that the gibberish lyrics were cries from a past life.
"Tea in the Sahara," which ends with three sisters dying of thirst in the desert.
"Mother". The theme may be innocuous (not wanting to get attached to a girlfriend), but Andy sings it like he's being tortured (which may be the whole point); that, plus the madhouse-like instrumental, makes for a truly nightmarish song.
"Friends", a song about a man who befriends people and then eats them.
There's also "Once Upon A Daydream," with copious amounts of Lyrical Dissonance. Perhaps not coincidentally, the previous three were all written or co-written by Andy Summers.
"Shadows in the Rain" a song about schizophrenia (or heroin addiction) which is pretty much made of Fridge Horror. Andy's atonal guitar noises in the background add to the creepiness.
"Invisible Sun", about The Troubles in Ireland, juxtaposes haunting synthesizers with lyrics about torture and murder.
"Behind My Camel" is a pretty sinister instrumental. It even won them a Grammy.
"Don't Stand so Close to Me" runs along the seduction of the innocent lines. Though the girl is almost an adult, the taboo of a teacher-student relationship sparks creepy feelings, and if you're a teacher, it would probably creep you out how relentless the girl is, waiting in the rain at the bus stop so he'll pick her up. How the temptation and frustration are "so bad it makes him cry" and his coworkers harass him "the accusations fly" . On the other side of the coin, the girl is tormented by her jealous friends and cruel classmates. It makes this more than a taboo, it's also a nightmare for both of the people involved.
Worse; about two decades later, a gender-swapped version of this very thing started to become incredibly common...
"A Kind of Loving" consists of a woman screaming throughout the entire song. Andy and Sting don't help by making atonal noises with their instruments. Recorded for a movie soundtrack and mostly forgotten about, it was included on Message In A Box, where it stands in sharp contrast with the other tracks.
Worse: the film, which Sting starred in, has a scene where his character rapes a disabled woman.
For the curious, the film was Brimstone and Treacle, written by Dennis Potter, who also wrote the great late 80s BBC drama The Singing Detective.