Which scales should I play over Beatrice Jazz standard?

I hear this song as mostly Lydian sounds - GbMaj7#4, EbMaj7#4, BbMaj7#4 mixed with a bunch of modes of the F major scale -

a couple of exceptions and other choices possible but mostly a Lydian tune -

Pretty inside sounding once you get the Lydian sound firmly in mind - but keep it simple and focus on the Lydian chord/scales and the F major scale -

In a nutshell - keep it simple and play the parent scales of the Lydian chords as you wish and then play F major scale over the F major scale modes -

My thoughts anyway -

FMaj7 chord - F Ionian (F parent scale)

GbMaj7#4 - Gb Lydian mode (Db parent scale)

FMaj7 chord - F Ionian (F parent scale)

EbMaj7#4 - Eb Lydian mode (Bb parent scale)

Dmin7 - D Aeolian mode (F parent scale)

EbMaj7#4 - Eb Lydian mode (Bb parent scale)

Dmin7 - D Aeolian mode (F parent scale)

Bbmin7 - Bb Melodic or Natural minor mode(possible Db parent)

Amin7 - A Phrygian mode (F parent scale)

Bbmaj7 - Bb Lydian mode (F parent scale)

Emin7b5 A7#9-quick 2 - 5 to D minor

Dmin7 - D Aeolian mode (F parent scale)

Gmin7 - G Dorian mode (F parent scale)

GbMaj7#4 - Gb Lydian mode (Db parent scale)

Fmin7 - F Dorian mode (Eb parent scale)

GbMaj7#4 - Gb Lydian mode (Db parent scale)

FMaj7 - F Ionian (F parent scale)

One way of analyzing this is to use the chord qualities to decide what modes/scales to use. The problem at hand is to decide what is considered a key change in the tune and what is not. I suggest this is a starting point:

Fmaj7 → F Ionian

Gbmaj7#11 → Gb Lydian (due to the #11)

Ebmaj7#11 → Eb Lydian (due to the #11)

Dmin7 → F Aeolian

Ebmaj7 → Eb Lydian, to retain the sense of F major, alternatively Eb Ionian if the Ebmaj7 is considered a key change. The difference between the two is the 5th tone, Ab (Ionian) or A (Lydian). I'd try the Lydian, since the 5th step in Eb Lydian coincide with the third of F major.

Bbmin7 → Ab Dorian/G Phrygian/Db Aeolian, what key should Bbm7 be considered a part of)

Amin7 → F Phrygian (retaining the tonal center of F major)

Emin7b5 → F Locrian

A7#9 → D mixolydian, but beware of the B, since the chord specifies a #9, C)

Dm7 → F Aeolian

Gmin7 → F Dorian

Gbmaj7 → Gb Lydian

Fmin7 → Eb Dorian, Db phrygian or Ab Aeolian, I'd start with Eb dorian, since this follows a temporary key change to Eb, being close to the key of F

Gbmaj7 → Gb Lydian

If we are strict about the qualities of chords, you should end up with these modes for the different qualities: Maj7: Ionian, Lydian

7: Mixolydian

m7: Dorian, Phrygian or Aeolian

m7b5: Locrian

In order to find the scales for maj7's and m7's, you need to find (or choose, or regard) the function for the chord. If you start with Fmaj7, you should use Ionian, since it is obvious that the tune is in F major. The greatest ambiguity is found in the m7 chords, since these can use three scales (based on chord tones found in the different scales).

Finally, you can play a lot of different things on 7th chords, in particular on altered 7ths...

I hope this was helpful! Scales are hugely useful to improvisation, and you really should study them.

If anything is unclear, make comments, and I'll try to explain.

The key, generally, is to make sure that the scale(s) you pick contain the chord tones as you go through the changes. The non-chord tones are less important, but may be influenced by surrounding chords.

To keep it stylistically modal, the simplest solution is to alternate between F major and minor pentatonics:

Fmaj7     → F maj pent  
Gbmaj7#11 → F min pent  
Ebmaj7#11 → F maj pent (but see Ebmaj7, below)
Dmin7     → F maj pent  
Ebmaj7    → F min pent (note: different from Ebmaj#11 because of Ab vs. A)  
Bbmin7    → F min pent  
Amin7     → F maj pent  
Emin7b5   → F maj pent  
A7#9      → F maj pent (be careful here, because the scale is a bit of a stretch)  
Gmin7     → F maj pent  
Fmin7     → F min pent  

Which pentatonic scale is determined primarily by Ab's presence (F min pent) or absence (F maj pent)

Another possibility is to move between various F modes:

Fmaj7     → F major  
Gbmaj7#11 → F phrygian (F Gb Ab Bb C Db Eb F) (i.e., Db major, starting on F)  
Ebmaj7#11 → F mixolydian (F G A Bb C D Eb F) (i.e., Bb major, starting on F)  
Dmin7     → F major  
Ebmaj7    → F mixolydian  
Bbmin7    → F nat. minor (F G Ab Bb C Db Eb F)  
Amin7     → F major  
Emin7b5   → F major  
A7#9      → F major (this is a stretch; second mode of D har. min works
                     better [ E F G A Bb C# D], but since the chord
                     doesn't last long, you should be okay as long as you're
                     mindful in your note selection.)  
Gmin7     → F major  
Fmin7     → F phrygian (because of the surrounding presence of Gb chords)  

In this case, the presence/absence of Ab and Eb in the chord has the biggest influence.

Finally, were this a song where hitting each chord change were important, then the rule of thumb is:

maj7    → major  
maj7#11 → lydian (major with #4)  
min7    → dorian (minor with nat. 6 rather than b6)  
min7b5  → locrian  
7#9     → dorian or mixolydian  

You could throw those in for color.