Can't stop at 10. These aren't in order or anything:

Hey, Jude — The Beatles
Sweet Soul Music — Arthur Conley
Summertime Blues — Eddie Cochran
Alabam — Cowboy Copas
After Midnight — Patsy Cline
Linus and Lucy — Vince Guaraldi Trio
Good Vibrations — The Beach Boys
I Can't Help Myself — The Four Tops
It Takes a Lot To Laugh, It Takes a Train To Cry – Bob Dylan
(I Can't Get No) Satisfaction — The Rolling Stones
Happy Jack — The Who
Glad All Over — The Dave Clark 5
Tchaikovsky's 1812 Overture — Stowkowski
How High the Moon — Ella Fitzgerald
I'm Gonna Be (500 Miles) — The Proclaimers
Pretty Woman — Roy Orbison
Biscuits in the Oven — Raffi
End of the Line — The Travelin' Willburys
Nine Pound Hammer — Merle Travis
Under the Boardwalk — The Drifters
My Girl — The Temptations
Honky Tonk Woman – The Rolling Stones
Hey, Good Lookin' — Hank Williams
Waltzing Matilda — Rolf Harris
Java Jive — Manhattan Transfer
Funeral March of a Marionette — Charles Gounod
Singing the Blues — Marty Robbins
MTD — The Kingston Trio
Black Diamond Bay — Bob Dylan
Oh, I Love You So – Preston Smith
King of the Road – Roger Miller
Sugar, Sugar – The Archies
Come Up and See Me – Steve Harley and Cockney Rebel
In the Mood — Glenn Miller
Love Me, Do – The Beatles
Big River – Johnny Cash
Sympathy for the Devil – The Rolling Stones
Norwegian Wood (This Bird Has Flown) The Beatles
Mr. Tambourine Man – Bob Dylan
Twist and Shout – The Beatles
I’m So Lonesome I Could Cry – Hank Williams
Sing Sing Sing — Benny Goodman
Folsom Prison Blues – Johnny Cash
Granny Woncha Smoke Some Marijuana – John Hartford
Copper Line – James Taylor
Shelter from the Storm – Bob Dylan
Johnny B. Goode – Chuck Berry
Rhapsody in Blue (Gershwin) – George Szell and Cleveland Orch.
Crying Time – Buck Owens
Handle with Care – Travelin’ Willburys
Trouble and Me – Buck Owens.

Billboard began to list the most popular recordings on January 3, 1938 (beginning with a Record Buying Guide, which eventually expanded into Juke Box Charts and Disc Jockey Charts, most of which were Top 10 or Top 15), but Joel Whitburn was only able to compile Top 20 lists for 1940-1948, and Top 40 lists for 1949-1954. The Hot 100 did not begin until 4 August 1958, but Whitburn was able to develop Top 100 lists for 1955-present.

Several major jazz artists had Billboard hits, especially in the early years, when swing bands were very popular. Among the many swing bands that frequented the charts were Benny Goodman (38 top 40 hits), Jimmy Dorsey (55 top 40 hits), Tommy Dorsey (68 top 40 hits), and Count Basie (12 top 40 hits), and Duke Ellington (9 top 40 hits), to name only a few of more than 100. Each of these artists had additional top 40 hits prior to 1940. Armstrong, for example, first made the charts in 1926, Ellington in 1927, Goodman in 1931, the Dorseys in 1935, and Basie in 1937.

Many legendary jazz artists also had top 40 hits, such as Louis Armstrong (14 top 40 hits), Ella Fitzgerald (more than 20 top 40 hits, including Mack the Knife (#27 in 1960), and more hits dating back to 1936), Sarah Vaughan (22 top 40 hits), and Dave Brubeck (Take Five, #25 in 1961).

During the 1940s, and probably earlier, jazz was so mainstream and popular that the number of jazz musicians with top 40 recordings is too numerous to mention in a short article.

More recently, many musicians who would consider themselves jazz artists, such as Norah Jones, have reached the top 40, and jazz is a major influence, if not the sole genre, of much of the popular music that climbs the charts. Whether it is currently waxing or waning in its influence is a more difficult question.

My top 10 list would be a mix of 80's retro and early to mid 90's alternative music they are in no particular order:

1.) Always Take The Weather by Crowded House (80's retro)

2.) Something To Say by Toad the Wet Sprocket (90's alternative)

3.) 100 Years by Five For Fighting (90's-05 alternative scene)

4.) I'll Stand By You by The Pretenders featuring Chrissy Hynde (80's retro)

5.) All I want is You by U2 (this song is classed as 90's alternative but is really just a love song)

6.) Angel of Harlem by U2 (see classification for#5 it's the same but it was written as a tribute to jazz singers Ella Fitzgerald and Billie Holliday)

7.) Smog Moon by Matthew Sweet (90's alternative)

8.) Between a Laugh and a Tear by John Mellencamp (80's retro)

9.) Heroes by Chad Kroger and Nickelback featuring Josie Scott of Saliva (00's rock it was on the Spider Man 1 soundtrack)

10.) Fallen Angels by Aerosmith (don;t know what genre you'd classify this song in but it's definately worth a listen

Honorable mentions would go to if it was only 10 tracks:

Nevada California by The Jayhawks

Walk Like a Man by Bruce Springsteen from the Tunnel of Love album

Talkin to My Angel by Melissa Ethridge (yeah I know she's a lesbian but I'm not prejudiced against gays and lesbians and I like her music and aint afraid to say it)

Runaway Train by Soul Asylum (great song and was used in a commericial for the National Runaway and Exploited Children hotline in the early to mid 90's)