Jeez, not a lot of love for the "classics" here. ;-)
I'm a big-band and swing guy myself, but I enjoy a lot of "modern" jazz that is tuneful and swings. Here are what are considered some pretty classic albums and some of my favorites:
Dave Brubeck - Time Out
Jimmy Giuffre - The Jimmy Giuffre 3
John Lewis - Grand Encounter
Charles Mingus - Blues & Roots
Duke Ellington - The Great Paris Concert
Rahsaan Roland Kirk - I Talk With the Spirits
Stan Getz and Jimmy Rowles - The Peacocks
Bill Evans - Sunday at the Village Vanguard
Ben Webster - At the Renaissance
Oscar Peterson - Girl Talk (Exclusively for My Friends)
Gerry Mulligan Meets Ben Webster
Sarah Vaughn - Sarah Vaughn (with Clifford Brown, 1954)
Dinah Washington - Dinah Jams
Billie Holiday - Recital (Verve, 1952)
That's just off the top of my head.