I am a vocalist trying to learn jazz improvisation without a teacher. I have been working on the modal scales. I am trying to cut through this dense forrest and I have a question: If I sing a song in A flat, can I pick any modal scale (Lydian, Dorian, etc.) starting from the A♭ note to compose a solo? Does this make sense?
Short answer is 'no'.
The mode you choose depends on the chord, and on the modal / tonal nature of the tune, and of course the musical context.
the tune is tonal, and is in Ab major. You should use the Ab Ionian mode on the AbMaj7 or Ab6 chords
the tune is tonal and is in Ab minor. You should use one of Ab aeolian / Ab ascending minor / Ab harmonic minor mode on Abmin7 / AbminMaj7 / Abmin6 chords
the tune is modal, and uses Ab , then you should use the indicated mode
Longer answer : you may play what you want, as long as you can explain what you are doing and why, and as long as it sounds as you want it to sound.
The musical context matters. The above indications are only a default sensible choice which will sound correct to most ears in a mainstream jazz context. You can choose to move away from these to get a fresh sound or sound external to the normal mode. For instance in a modal tune using Fmin7, it can be perfectly ok to use Gb and A. A typical example of this is the main theme of Beirut by Steps Ahead () and Mike Mainieri's wonderful vibraphone solo on the same tune (transcription available here). But using these kind of notes in a traditional New Orleans context is not appropriate.
Starting from Ab. All the modes can do that. Except each has a different flavour. Ab Ionian will give a total major diatonic sound, such as most pop type songs use. Ab Aeolian will sound minor, but without the notes that melodic and harmonic have at their top ends. Ab Dorian will also sound minor, but again will sound different at the higher end (of its scale). Ab Mixolydian will have a major feel, with a blues tinge at the top end, and Ab Lydian will be major, but one note will sound out of place , as it's not in the major diatonic scale. Ab Phrygian will have a sort of Spanish/Arabic feel to it, in a minor way. Then there's Ab Locrian...
There's also the modes of Ab minor, which opens up a whole new can of worms.
So, of course you can, but need to be aware of what else is happening - chord sequences will dictate a lot. Whichever you choose, coming back to that Ab note a lot is important, as that will be your 'home base', otherwise the structure could revert to the parent key feeling like home. As in Ab Dorian can easily start to sound like F#/Gb major.
I'd offer that the best way to learn jazz improvisation is from the recordings. For modal jazz, the most famous example is also a great place to start: Miles Davis' solo on the tune "So What".
"So What" has two chords: D minor, and Eb, both with a flatted seventh. Listen to Miles' solo on the recording, and learn to sing along with it. What notes and scales does he use? Which notes does he linger on, and which are passing tones?
Once you've identified the basic scales he uses, listen for where he deviates from these, and includes notes that aren't in the scale. How does he do this? Are any of the notes emphasized, or does he use them as 'passing' tones?
While scales are important to know, and become important when considering how to improvise on longer passages of only a few chords, it's also important to remember that most jazz musicians don't consciously think about scales while they are improvising.