Al Johnson - ‘Way down yonder in New Orleans’
Actually Al Jolson if you want to listen.
The Andrews Sisters recording of Bei Mir Bistu Shein. It’s easy to hear how it was such a hit, and for me, it evokes images of some of the best elements of that era. Other big favorites already mentioned in this thread are “In the Mood”, “Boogie Woogie Bugle Boy”, and “Fever”. I’m also fond of “A Kiss to Build a Dream On” (and, in fact, nearly everything else by Louis Armstrong) and The Ghost of Smokey Joe.
I’m not going to get into the really old music I like.
“In the Mood” and “Sing Sing Sing” are very good choices. Personally I have a very soft spot for “Shout Sister Shout” by Sister Rosetta Tharpe and Lucky Millinder (YouTube Spotify) as I loved dancing to it when I started learning to Lindy Hop. Then there’s “Way Back Home” by Bob Crosby and the Bobcats (YouTube Spotify) after it was used in “Fallout 3”.
Walter Houston September Song
A few of my favorites from my mother’s stack of 78s that I used to play when I was a kid:
Hal Derwin - The Old Lamplighter
The first ones that occur to me:
Sweet Leilani by Bing
I Don’t Want to Set the World on Fire by The Ink Spots
Huggin’ and Chalkin’ by Hoagie Carmichael 'cos it makes me laugh
My favorite is “Goodbye” from Benny Goodman. My parents used to play a lot of big band stuff when I was a kid, and most of it didn’t stick with me but I’ve always liked this one.
For crooners, another from my parents’ collection: “More,” the theme from “Mondo Cane.” It was my parents’ song for 47 years, and every time I hear it these days it makes me a little wistful (Mom died last year).
Also, Frank Sinatra’s “It Was a Very Good Year.”
Apparently I like old-fashioned downer music. 
(Looking at the OP again, those last two might be too new–they qualify as crooners, but they’re from the 60s.)
In the Mood has been a favorite of mine for a long time.
I’m also a big Les Paul fan, and his version of How High the Moon, with Mary Ford, is awesome (even more so when you know about the amazing multitracking, for that era, which they did to achieve the sound).
I like this one more because it has a good association for me than for itself. It’s not on my regular playlist, but I smile when I hear it.
You can find them all here (I favor the '50s):
Wow, that is a very nice site, and the recordings seem to be in very good shape. Thanks for the link!
Besides the already mentioned Begin The Beguine, other Artie Shaw favorites, Frenesi, Dancing In The Dark, Moonglow and Stardust.
After reading the whole thread I even searched to be sure I didn’t miss it - guess I get the Ink Spots all to myself! Java Jive brings back some of the most powerful nostalgia of my life. But ‘I don’t want to set the world on fire’ is one of my all time favorites to sing along to.
I loved singing their songs so much, the Ink Spots ruined some of my classic choral training. Tough nuts. Their syncopation is better than anything Robert Shaw could come up with.
Peggy Lee singing Rodgers and Hart’s Lover.
I have “Sing Sing Sing” as my ringtone. But I’m also quite fond of “Begin the Beguine.”
My favorite big band song changes day to day, but Tuxedo Junction by Glenn Miller is right up there.
There’s a version of “Tangerine” that I adore but can’t find. I can’t think of the singer, which is a problem.
“The Girl from Ipanema” isn’t that old, but I adore it.
As for crooners, though, probably my favorite song: “Stardust” sung by Nat King Cole. The combination of that voice and that magnificent song is just perfect.
We’ll Meet Again was, for my grandmother and my grandpa (her second husband, not my biological grandfather)“their song.” He was a sailor on the Great Lakes, and they dated on and off for about 6 years before they eventually married, at the end of shipping season in 1954. I listened to Vera Lynn and songs of that era at their place, and later working in long term care and with seniors, I got to know others… the Andrew Sisters “Don’t Sit under the Apple Tree” is one of my favourites to sing a long to. Glenn Miller is fun, brings a smile to everyone of a certain age (and also the next generation, their children).
The day after my grandmother died, when I was shopping for something to wear to her funeral, (I wore a grey suit that I already owned, but I bought a pink blouse…she liked me in pink) I found a bargin bin with Benny Goodman, Vera Lyn, Andrew Sisters, and Glen Miller cds. I bought about half a dozen, and played a mix of those when we had people over. Everyone commented on the music, although only one great uncle remembered these songs as being on the hit parade.
Now I usually loan them out to recreationists to put on tape or mp3 player for people in our care. I don’t play them often, but good memories. Vera Lynn gets played every Nov 11th, and also December 18th, my grandparent’s anniversary.
From my own experience it was Perry Como and Andy Williams. I thought they were “dreamy.”
Awww, I’m sorry OP. :smack: Only Big Band? Then I’d have to go with Locomotion. 