992: “We came a long time ago, 9-9-2 but you did not know” - America: What Time Is Love? by The KLF
1969: “In 1969, I killed a man of mine” - Tomb Of Liegia by Team Sleep
1985: “In 1985 I was doing time alive” - Tomb Of Liegia by Team Sleep
1995: “1995 was the year I came up for trial” - Tomb Of Liegia by Team Sleep
Similar to this, I’ll add some Pogues:
1841. “In eighteen hundred and forty-one, my corduroy britches I put on” - Poor Paddy by the Pogues.
1842. “In eighteen hundred and forty-two, from Hartlepool I moved to Crewe” - Poor Paddy by the Pogues.
1843. “In eighteen hundred and forty-three, I broke me shovel across my knee” - Poor Paddy by the Pogues.
1844. “In eighteen hundred and forty-four, I landed on that Liverpool shore” - Poor Paddy by the Pogues.
1845. “In eighteen hundred and forty-five, when Danny O’Connell he was alive” - Poor Paddy by the Pogues.
1846. “In eighteen hundred and forty-six, I changed my trade to carrying bricks” - Poor Paddy by the Pogues.
1847. “In eighteen hundred and forty-seven, poor Paddy was thinking of going to heaven” - Poor Paddy by the Pogues.
And a few more:
1846. “In eighteen hundred and forty-six, on March, the eighteenth day” - Greenland Whale Fisheries by the Pogues.
1915. “In 1915, my country said son, it’s time to stop rambling, there’s work to be done” - And the Band Played Waltzing Matilda by the Pogues.
1953. “On Canvey Island, 1953, many lives were lost” - Canvey Island by British Sea Power.
1959. “Can you hear that '59 sound, coming through my grandfather’s radio” - The '59 Sound by the Gaslight Anthem.
1982. “And now you find yourself in '82” - Heat of the Moment by Asia.
I’m trying to figure out why my suggestion for 1981 (post #22) didn’t make the cut.
2005: “And if you survive till 2005, I hope you are exceedingly thin” Pink Floyd (single: Point Me At The Sky)
and it goes on: “For if you are stout
You will have to breathe out
While those around you breathe in.”
I love that. Very prescient.
I listed 1968 from Dave Alvin earlier, but he’s got a few more:
1861: “In the spring of '61 I kissed my mother goodbye” Andersonville by Dave Alvin
1962: “This song comes from 1962 dedicated to a man who’s gone” Border Radio by Dave Alvin
1955: “And he closed his eyes and hit the stage in 1955” Haley’s Comet by Dave Alvin
1963: “They paroled Blue Wing in August, 1963” Blue Wing by Dave Alvin
1959: “… when I was young and in my prime back in Gary Indiana in 1959” Gary Indiana, 1959 by Dave Alvin
1849: “…dwelt a miner, 49er, and his daughter Clementine.” Oh My Darling Clementine, Percy Montrose
1989. “Haven’t been as high as this since the night I burned my lips. Must have been, like, '89.” Mission Viejo by Lifter Puller
1607. “In sixteen hundred seven we sailed the open sea.” - “Virginia Company” from Pocahontas soundtrack
1952. “Its a Vincent Black Lightning 1952” - “1952 Vincent Black lightning” by Richard Thompson
1953. “Ain’t had no money in my pockets since way back in '53” - “Alimony” by Ry Cooder
1492. “In 1492 Columbus sailed the ocean blue” – 1492 by Counting Crows.
1493. “In 1493 he came home across the deep blue sea” – 1492 by Counting Crows.
1494. “*In 1494 he did it with the girl next door *” – 1492 by Counting Crows.
1495. “In 1495 he barely made it out alive” – 1492 by Counting Crows.
1964. “*In 1964 ?? busting out the door *” – 1492 by Counting Crows.
1970. “*In 1970 some people got their hands on me *” – 1492 by Counting Crows.
1492. “*If you were Chris Columbus Back in 1492 *” – 1492 by Betty Johnson.
1492. “When we take it, we’re going to make it 1492, 1492” – Ghost Dance 1492 by Beachwood Sparks.
1492. “In the year of 1492 When Columbus sailed the ocean blue” – Galileo by Amy Grant.
1492. “Like the first man on the moon, Columbus 1492” – Uncharted Territory by Rachael Lampa.
1492. “In 1492 just to see what he could see” – Listen Mr. Bilbo by Pete Seeger.
1988. “And I met William Butler Yeats, Sunday Night Dance Party, summer, 1988.” - Chicago Seemed Tired Last Night by The Hold Steady
1883. “Well I found myself in Griffin, in 1883” - Buffalo Skinners by Woody Guthrie
(Note: This is a traditional song and various versions use other years, including 1873 and 1833)
1956. “I remember you in Hemlock Road in 1956” - Memo from Turner by Mick Jagger
1962. “This song comes from 1962” - Border Radio by the Blasters
1963. “It sounds like 1963 but now it sounds like heaven” - Windfall by Son Volt
1965. “Weren’t you at the Coke convention back in 1965” - Memo from Turner by Mick Jagger
1949: “In nineteen and fifty-nine we took a little hike” - The Battle of Kookamonga by Homer & Jethro
Ah, I should have thought of that one. Great song. 
1993. “Yeah, this is how we chill from 93 'til” - '93 Til Infinity by Souls of Mischief
1945. They disembarked in 45- Southampton Dock by Pink Floyd.
1918. “We taught them a lesson in 1918 and they’ve hardly bothered us since then” – MLF Lullaby by Tom Lehrer
1949 James Dean in that Mercury '49 - “Cadillac Ranch” - Bruce Springsteen
1969 I got a '69 Chevy with a 396 - “Racing in the Street” - Bruce Springsteen
1964: All alone at the '64 World’s Fair/Eighty dolls yelling “Small girl after all” - “Ana Ng” They Might Be Giants
1963 I was born in '63/Got a little job at the factory “The Factory” Warren Zevon