Harry Chapin’s Dance Band on the Titanic fits the OP - even if it’s hugely innaccurate.
Then there’s Johnny Horton’s Sink the Bismark!.
A bit harder to find is the Civil War ballad, Roll, Alabama, Roll about the Confederate Raider Alabama.
One more - I had to check to make sure it was based on real events, though:
The 1938 NewYork mining disaster by the BeeGees.
Another great Wooddy Gutherie song is about the USS Ruben James
Each time that song is mentioned in this thread there’s a new spelling. 
Oooops. But I added a link. That must count for something.
Blind Willie Johnson’s Blues song about Titanic, God Moves on the Water.
Wreck on the Highway, written by Dorsey Dixon about a car accident in North Carolina. Later recorded by Roy Acuff.
A reference, of course, to Grateful Dead’s take on the Casey Jones legend.
Nitpick: The song’s title is “New York Mining Disaster 1941.”
He’s not well known outside Canada, but Stompin’ Tom Connors has a couple of disaster songs:
“The Bridge Came Tumbling Down” is a song about the collapse of the Second Narrows Bridge in Vancouver in 1958. Nineteen ironworkers lost their lives; the official name of the bridge was changed to the “Ironworkers Memorial Bridge.”
“The Martin Hartwell Story” was about the crash of a small plane in the Arctic in 1972, the search for it, and how the lone survivor managed to keep alive.
There’s a hard-to find old gospel song about the 1946 burning of the Winecoff Hotel in Atlanta, the worst hotel fire in US history (119 dead, including a large number of the state’s brightest high school students, in town for a mock legislature). One haunting lyric I still remember from hearing the song on an old 78:
Those faces in the windows
I see them in my dreams
I always will remember
Those agonizing screams
The song itself is nothing about any sort of disaster - the lyrics are actually fairly insipid stuff about a lost love. Such is life when you have to write words to fit into an existing melody that someone else put together.
From http://en.wikipedia.org/wiki/My_Heart_Will_Go_On:
*[composer James] Horner had originally composed the song as an instrumental motif that is used in several scenes during Titanic. He then wanted to make a full vocal song out of it, for use in the end credits of the film. Director James Cameron did not want such a song, but Horner went ahead anyway and got [songwriter Will] Jennings to write the lyrics. *
Of course, I think the song itself is a trainwreck, so it becomes a circular reference. 
“Just sit right back and you’ll hear a tale,
A tale of a fateful trip …”
What?
Hey, it was real to ME.
The Titanic had an entire musical
And that great children’s ditty:
They built a ship Titanic
To sail the ocean blue.
And they thought they had a ship
That the water wouldn’t go through
But the Titanic didn’t make it
Across the ocean blue.
It was sad when the great ship went down.
A note about the Rueben, Reuben, Ruban, etc. James. I have an old Pete Seegar album which includes this song. One of the recurring lines from the song is “Oh, tell me what were their names, what were their names? Did you have a friend on the good Rueben, Ruban, Reuben, etc. James?” On the back of the album cover is a list of the names of the sailors who went down with the ship.
On Reckoning/For the Faithful, Bob Weir describes The Monkey and the Engineer as a song about “tragedy narrowly avoided.” But for the quick thinking monkey, this song would fit in this thread perfectly.
ETA: Ooops. Just re-read the thread title. I don’t think this one was based on a real event!
Another Due South fan here. Apparently, they’d planned to use the real Edmund Fitzgerald as the “ghost ship”–along with Gordon Lightfoot’s wonderful song. But interviews with survivors convinced them that it would be too painful. So Paul Gross got to write a new song (which I’m sure he hated). However, the link to the Fizgerald could still be felt in the 2-episode story. Add in Mounties sailing a replica of the HMS Bounty, a daylight “romantic dream” sequence, an environmental message, silliness from various supporting characters & a kiss between Fraser & Kowalski. (Yeah, “Buddy breathing” involves the tongue!)
Did I say I was a fan?
My grandma used to tell us about a train wreck back around the turn of the last century. Her uncle was an engineer who realized that a mistake had been made–his train was going to crash into another one. So he stayed on the brake until the last minute, hoping to reduce the damage. And was thrown into the firebox.
She said there was a song–but she died in 1966. Just another detail I never learned. (Hmmm… a research topic?)
Townes van Zandt covered “The Wreck of the Old 97.” Here’s a the story. (Or maybe it’s just one version–that’s “the folk process” for you.) And Thomas Hart Benton’s painting.
Would “Ameria Earhart’s Last Flight” count?
And there’s no record of any mining disatster in New York nor anywhere else in the US in 1941.
Sally Ride was on Challenger but not during the disaster. As far as I know, she’s alive and well. I think you’re thinking about Christa McAuliffe. So you can go back to enjoying the song again.
Oops, my bad. Coffee first, then posting.
If we are including mining disasters, there should be a mention of Timothy, except it’s not real-life (I hope)