Rock a Bye Baby: "Rock a bye baby, on the tree top. When the wind blows, the cradle will rock. When the bough breaks, the cradle will fall. And down will come baby, cradle and all." Seriously, look at it. It's a song about a baby falling off a tree, although there are alternative theories about this being a "dandling song" sung whilst the baby is being tossed in the air or that it portrays the baby's nightmare whilst being rocked in its sleep.
Also Lampshaded in one of the original Simpsons shorts, where Marge sings this song to Maggie, who has a nightmare about falling out of a tree during the song.
Likewise, the German nursery rhyme Hoppe Hoppe Reiter, which loosely translates to "Hop hop goes the rider, when he falls he screams. If he falls in the ditch he'll be eaten by ravens, and if he falls in the swamp he'll go splash." Other versions translate ditch as grave, and add a couple lines about being bitten all over by mosquitoes. This is a song that's used to entertain children. Oh, those cheerful Germans. The rhyme accompanies a game with a small child as the rider and the parent's knees moving up and down as the horse. At the end, the knees move apart, but the parent prevents an actual fall.
Waltzing Matilda does this pretty bloody well. It's a song that every Australian is taught as a child, and therefor in every Australian's life there is that little Oh, Crap moment too when they first fully appreciate the lyrics...
The children's song "Ring Around the Rosie" is actually about the millions and millions of people dying during the Black Plague. The posies were lain as burial flowers, and then the bodies of all those who "fell down" were burned into "ashes."
Most folklorists believe this is incorrect, the apparent plague references are absent in the oldest versions of the song which is itself not very old, the first published versions date to the 1880s with some references to it from the 1850s.
The French children's song "Alouette" is about plucking (i.e. ripping the feathers out of) a lark. Specifically, it lists the different bits of the bird that the singer will pluck in order. Including its beak.
The Christmas carol "Do You Hear What I Hear?" seems like a cookie cutter religious song about the birth of Jesus and how word got to the king. However, the truth of the matter was that the song was written in 1962 as a plea for peace during the Cuban Missile Crisis. That makes the first verse of this song extremely horrifying when you realize that they were likely referring to a nuclear bomb.
Not just that, but who was the king during Jesus' birth? King Herod, the one who killed every single baby just to try to get to Jesus! Looks like it wasn't the wise men who told on him after all, but an innocent little boy.
In the Herman's Hermits song I'm Henry VIII I Am, he's talking about a widow who's had seven previous husbands (all named Henry). If she's been widowed seven times and still looking for another husband, it's clear this marriage won't last long—and considering she's a widow, neither will the singer...
The band Zebra, has a song "Who's Behind The Door." Not only do are the people at the computers in the music video strongly similar to government types but the lyrics have very heavy themes of both existential surveillance (i.e. God or maybe an Eldritch Abomination is watching) and mundane surveillance (i.e. the CIA or FBI is also watching). But don't panic.
They watch us all They're only making sure that we don't trip and fall/ Now they look so hard but they can't tell us why they're here and just what for/ Because they don't know Who opened up the door/ How can we find out more Who owns the keyless door...
"You are my Sunshine is a classic example of a Chorus-Only Song. People who have heard the full version will often tell you it's not about a parent and their child, but a person whose lover has broken up with them. The song however can be intepreted as a song about a parent whose young child has died. The lines "but if you leave me to love another, you'll regret it all one day" sort of ruin both interpretations unless you take it as a song about a very overprotective parent.
Maybe more Fridge Squick than Fridge Horror, but in Kenny Rogers' "Coward of the County", he sings about how the Gatlin boys "took turns with Becky", and then adds - in a very matter-of-fact way - "there was three of them". Think hard enough, and then you realize that they may not have just taken turns with Becky, but may have taken advantage of her all at once.