From Mark Levine's The Jazz Theory Book:
The Jazz Theory Book
We've now completed three of the four scales from which most of the chords played by jazz musicians are derived [Major, Melodic Minor, Diminished]. There's only one to go and it's the simplest, and least played of the four scales: the whole tone scale.
Further on a bit:
You've now learned about all four of the scales [Major, Melodic Minor, Diminished, Whole Tone] you'll need under your fingers to play over chord changes.
So, from one of the most respected and well known sources for jazz theory the answer is:
Major: Ionian mode.
Melodic Minor: Levine describes this as a major scale with flatted 3rd - using C Major, Eb instead of E. This seemingly small difference has great ramifications when developing melodic minor harmony.
Diminished: A symmetrical 8 note scale with two forms - Half step/Whole step; Whole step/Half Step.
Whole Tone Scale: A symmetrical 6 note scale, every interval comprised of a whole step (two notes in the chromatic octave), resulting a #4 and a #5. Here's a whole tone octave starting with A: A, B, C#, D#, F, G, A. (There will always be one note missing when spelling out a whole tone scale - where to put it is discretionary - in this case I skipped E, using F instead of what could be called E# - a common way of doing it.)
Important: The point is not simply to learn these scales. Levine devotes considerable space to discussing each of them, with all of their modes and derived chords, and how their various harmonies work and are used in jazz. (His discussions will lead you to many of the other scales and forms discussed in other answers posted here.) The modes and harmonies derived from these four simple scales encompass huge swaths of jazz music and jazz theory.
Familiarity with that material unlocks the secrets of good/great jazz playing to those dedicated and talented enough to penetrate it and play it. (That doesn't really include me...)
This part of the question was subsequently edited out, but I am leaving this section intact because the information is, IMO, valuable:
As a bonus question, should they be practiced with all the notes as
the tonic or just with "musically interesting" tonics with respect to
the considered scale ? If so, how to identify which tonics are the
most important ?
You are apparently asking which modes [see endnote] are the most important ones: Playing a mode of a given scale means using the same key signature as the 'parent scale' but starting the scale on a different note, which is called the "root". ("tonic" is a term which is usually used in reference to chord progressions or discreet melodies, where that note acts as the tonal center - think of "tonic" as "tonal center" - not with abstract scales, when it's called the "root" - the base/reference note on which the scale is built.) The most familiar example is Aeolian mode, what we call the Natural Minor, which is the 6th mode of the C Major scale: A scale with no sharps or flats, using A as the root.
Rather than venturing a guess about which modes are the most used, I'll again relate what Levine has to say on this: (Emphasis is his)
Remember your goal: to see, think, and play scales as an available pool of notes, of which "do-re-mi-fa-so-la-ti-do" is only one possible combination.
So in essence, all notes are created equal - they all can and should be used to make music, depending on the context and your own ear and talent, and that understanding is a cornerstone of modern jazz playing. (Atonal and "free' jazz are styles that flourished in the late 50's and 60's - not usually easy music to listen to, often looked down upon. But those artists were driving home some very important points - among them is this one: All notes are created equal.) Levine also devotes whole chapter to practice techniques that will help you realize that stated goal.
If you have a solid grounding in basic theory and want to get up and running with jazz, you will be hard pressed to do better than Levine's book. (If you're new to theory, it will be very rough going.)
The term mode is often the subject of confusion, but it's really
Mode in the musical sense means a 'manner' or 'fashion' of playing a scale. There is no absolute difference between a mode and a scale:
Every scale has a given key signature and one can play any ordered
collection of 7 notes using that key signature - start on any note in
the chromatic octave (which has no accidentals) and play a
conventional 7 note scale, following the key signature - you now
have a what's called a scale or mode.
Mode is a relative term: when you decide, for whatever reason, to make one particular note your root, that pattern is called the
scale, and becomes a 'parent scale', as it were, to its other modes: Building a scale with the same key signature but using a different root than the 'parent' root becomes a mode: A different
fashion/manner of playing the parent scale.
So using the example of C Ionian - which we call the C Major Scale,
the 6th mode of that scale - a scale that starts on the 6th degree of
the parent scale, (whose key signature has no sharps and no flats) in
this case A - becomes the Aeolian mode, commonly known as the A
Minor Scale. But note that C major also has a modal name: Ionian. In
fact, if we decide that A Aeolian is the parent scale, C Ionian would
then be considered the third mode of A Minor. And so it goes with
all modes and scales.
Because today we tend to think of the C Major and A Minor Scales as
the basis for our tonal system, everything else we refer to as
"modes", but it's by no means so clear-cut, and in truth it's simpler
than one might think.