When my wife was pregnant with our first she heard “Baby Mine” (aka, that one song from Dumbo) on the radio or a TV show or something and burst into tears. Five years later she sings it to the two of them probably weekly, when they’re having trouble going to sleep, there is a storm, etc. Music is such a wonderful thing.
(My wife too! But she’s always embarassed when I listen to her sing to the kids. Why? I hear her sing all the time.)
When Lil’ Lebeef was born I didn’t know any lullabyes, so I sang what I knew. I Wanna Be Sedated a la Rhett Miller, lots of Roger Miller and Tom T. Hall.
Now anytime I sing Ba Ba Ba Ba, a little two year old voice chimes in with “wanna be sedated”. It’s awesome!
I sang “Lullaby of Broadway” to my daughter every night.
You are my sunshine. It works particularly well because you can slow it down and it still sounds nice. I started singing it when she was in the womb with the idea that she’d recognise some of the rhythms when she was born. Perhaps it worked, or perhaps it just made me feel calm, which helped her; it wasn’t much effort either way.
Two of the others were my own made-up tune with the lyrics of two German poems by an acquaintance of mine. I wanted her to learn German (I failed) and they just worked so well as lullabies.
I also used to sing “hushabye mountain” from Chitty Chitty Bang-Bang.
Plus the usuals - rock-a-bye-baby, etc. But the ones listed above were the standards and You are my sunshine was the one that was so familiar she’d calm down even when she wasn’t calm.
Also not a parent, but a long-time singer of lullabies to friends’ children. My top four are:
“Distant Melody” from Peter Pan
“Back When I Could Fly” by Trout Fishing in America
“Lullaby” by Trout Fishing in America
“Tell Me Why” (no idea where it originated; I learned it from Pat Benetar on the original For Our Children album)
My 3 are “You Are My Sunshine,” a nonsense one I made up called “The Baby Sleeps” and “Song for the Mira.” By the time I’m to the last verse of that one, my 10-month-old is almost always asleep. It fits well - “sweet be your dreams, and your happiness swell.”
I forgot I Gave My Love A Cherry. One of the favorites as well.
“Away in a Manger” was a big hit with my son.
Its a long way to Tipperary.
I t was the only song that calmed our first born, she had to be swung in a hammock and sung this song, and was then a happy sleepy baby.
“Little Black Rain Cloud” from Winnie the Pooh and the Honey Tree for my son, and “World of My Own” from Alice in Wonderland for my daughter. I have a thing for Disney movies based on classic children’s books. 
All sung in a soft drone at slow pace. Longer songs work better.
“I’ve been working on the railroad…” – Each verse has a different tune/cadence; I repeat whatever one is most soothing.
“Good King Wenceslaus…” – Because of a quirk in my brain, I have all the verses memorized. Who says 7th-grade choir was a waste of time? 
“On the first day of Christmas…” – Great long song, especially if sung slowly.
Oh yeah, I forgot to mention my papa’s lullaby for us: Riders on the Storm, much to my mother’s dismay. 
We loved it! If my mum complained he always said he only knew the words to that and the Matthäus-Passion.
I’m a terrible singer and once my son figured that out (around age 2), he pretty much ordered me never to sing to him!
But when he was a baby, he liked it when I’d sing old Irish ballads.