There are probably 4 jazz waltzes that come up in jam sessions more than any others. They are Miles Davis' "All Blues," Richard Rodgers' "My Favorite Things," Toots Thielemans' "Bluesette," and this one: Freddie Hubbard's "Up Jumped Spring"  It is a lively, happy song that is true to the title.  Nothing terribly complicated in this one, other than learning to be comfortable in 3/4 time

This is a lesser-known song by Horace Silver: "Peace".  It is a beautiful, haunting melody that is a great platform for working on your ballad chops.  But don't be fooled by what sounds like a simple composition.  This has an unconventional song form and very unconventional chord progression.
It is a 10-bar form.
The chords don't go where you might expect them to go.  Here is the chord progression (in concert key).  Please notice that there are lots of half-diminished (i.e. m7b5) chords and also many 7(b9) chords which are essentially diminished chords with an altered bass, work on those diminished intervals:

||: Am7(b5)    D7(b9             |  Gm             C7        |
    Bmaj7        Cm7(b5)  F7   |  Bbmaj7                  |
    Bm             E7                    |  Amaj7       F#m     |
    Ebm7(b5)  Ab7(b9)           |  Dbmaj7                 |
    C7(#9)        B7(#4)             |  Bbmaj7                 :||

Here's an addictive little Sonny Rollins tune that every jazz player should know.  Nothing very complicated here.  Your basic 12-bar F blues.  The challenge here is the tempo, 190 BPM.  Get those fingers, tongue and brain working together.

Ellington's classic is an easy tune to chill out to without making your head hurt.

"Once I Loved" is one of Jobim's nicest Bossas, that is obscured by the more famous songs like "Girl from Ipanema" and "Wave".  Like all Jobim songs, it is brilliantly crafted with interesting harmonies and graceful melodies.
This is such a natural melody that is relatively easy to solo on this tune.  The only real complication is that it is a different song form than the usual 12-bar blues or 32-bar AABA or ABAB song.  I would call this song form AAB with a tag.  The A sections are 8 bars as usual.  The B section isn't really a "bridge" in the sense of connecting two parts of the song.  The B section simply begins the second half of the song.  The B runs 8 bars, which is natural enough.  Then the last 4 bars of that B section are played again (the "tag").  That gives us a chorus of 28 bars, not the typical 32.  It flows so naturally, the audience doesn't miss those 4 bars at all.  But the musician must know the difference.

How about some blue skies for a rainy day?  Here is a smooth jazz treatment of this classic.  Good for working on your phrasing and legato sound.

"Have You Met Miss Jones" by Rodgers & Hart has one of the most maddening bridge sections of any tune in the "American songbook."
The song is a straightforward A-A-B-A form of 32 bars.  The A section couldn't be more conventional, in the key of F concert, the A section has one chord per bar, all in that same key center.  To start with this tune, you can use notes from the F concert major scale throughout the A sections.  The changes are:
I  |  vi  |  ii  |  V  |  iii  |  vi  |  ii  |  V  |
There are thousands of pop, jazz and classical songs that follow a progression very similar to that.  So it seems pretty easy.  Until ...

Until you hit that bridge.  The bridge goes all over the place, completely disconnected from the F key center.  Many a good player has suffered a train wreck here.  To get through the bridge, you can break it down.  There are three sets of ii-V-I patterns.  More often than not, the ii-V-I pattern begins on an odd numbered measure, typically bars 7-8 of a section. In this case, each of the ii-V-I patterns begins on an even-numbered measure, 18-19, 20-21, and 22-23.  You can think of the ii-V as leading you to a new resting place.  So think ahead to those resting places (Gbmaj7, Dmaj7, and Gbmaj7.) This gives the bridge a sense of "leaning forward", which is a big part of the genius that made this song a standard.

And if you survive to the Gbmaj7 in bar 23, notice there is one more ii-V-I sequence (measures 24-25) taking you back to the key center of F for the final A section.  So in that bridge, everything seems to happen one measure before you think it might.

Here's a smokin' little Sonny Rollins tune that isn't played often enough.

Ready for something a bit more challenging, or at lease more uncomfortable for many of us?  Here is Wayne Shorter's "Footprints".  It is basically a modal piece that stays on Cm7 for almost the entire tune.  That's a challenge to craft a solo that is interesting when the harmony is not changing.  Experiment with playing different scales over that Cm7.  The melody line mostly follows the Bb major scale, for example.

But then there is that bridge -- 4 bars of very complex chords in bars 17-20 of the head: F#m7b5, F7#11, E7b5#9, and A7b5#9.  That's crazy.  Thing about whole tone scales, diminished arpeggios, augmented arpeggios.  These 4 bars go by quickly.  Attitude is probably more important than the actual notes

Oh, and did I mention it is is 3/4, just to make sure your head explodes?

This is a case where you might find it helpful to scat sing solos for about 4 hours before trying to do something on the horn.

The bridge is especially difficult because it doesn't follow any of the more familiar patterns, such as circle of 4ths or ii-V-I.  Shorter specified rather complex chords, but rhythm sections will not necessarily play it literally as written.  Here is a slowed-down version of the bridge and the chords are played in a way that is easier to hear.  You will trade 8s with a tenor sax that provides some good examples.  The loop is 8 bars:

Cm7        | Cm7        | Cm7        | Cm7        |  
F#m7b5  | F7#11      | E7b5#9   | A7b5#9  |

It is almost April Fool's Day.  How about a happy little jam track for "My Foolish Heart"?  This one has 2-bar lead-ins between choruses to practice your a capella lines.  There is also a vamp at the end with several open breaks.