The innovations of Charlie Parker and Dizzy Gillespie in the 1940s changed jazz forever. These two guys codified how the language should be spoken. This scale is one result of their efforts. This particular term for the scale was coined by David Baker, so I use it myself.
The Bebop Dominant Scale is basically a Mixolydian mode with an extra note — a natural 7th. Here is the basic C Mixolydian mode:

The Bebop Dominant has the extra natural 7th, like so:

This added note does two things: 1) Provides a smooth descending leading tone, 2) gives the scale 8 notes. Item 1 is immediately obvious. Item 2 is not apparent until you start soloing with the scale. What the 8 notes do is give you the ability to play chord tones on all downbeats. This is important; it is the most effective way to get long phrases. For example, check out this standard Mixolydian lick:
Notice how the last half starts to sound awkward. You don’t have chord tones on downbeats. Look at this bebop lick:
Do you hear it? It just wants to keep going. That’s how those guys like Parker, Gillespie, Adderley, etc. could get those endless lines. Even Pat Martino uses this technique. Remember this however: Use the added 7th ONLY when descending. Skip it when you ascend. And KEEP THE CHORDTONES ON DOWNBEATS! By the way, I highly recommend David Baker’s 3 volume set on how to play Bebop.
This little device can revolutionize your playing if you take time with it. It can change how you view improvisation. I am not kidding. The Bebop technique is one of the secrets of the masters. Take this one home and lock yourself in a room for a week. Experiment. Make your own licks. Create lines. Then, go get a gig!
Bebop lines over the Minor 7 Chord
Just as you can use them over their native Dominant 7 chord, Bebop scales can also be used over a Minor 7 chord whose root is a 4th below the root of the scale. In other words, anything that works over C7 will work over Gmin7. For your further practicing enjoyment, here are some minor licks in the Bebop style that I came across some years ago. They have a Pat Martino-ish sound to them. Enjoy!
Hope you liked those. You should transpose them in all keys so you can use them whenever the need arises.
Bebop scales over the min7b5 chord
This will be s short one, but the information will take a while for you to practice. This is a little technique that gives you more sounds over the half-diminished (or min7b5) chord. If you refer back to the earlier lessons you will notice how I have covered topics that all relate to each other. Here is another one. Up till now, the only scale choice I have discussed for this chord has been Locrian. It has been a useful chord, but it only takes you to the first level: the I chord (see lesson 2 for an explanation of this concept). This next scale takes you all the way to the level of the V – the most harmonically rich level.
Follw me here: The min7b5 is based on the vii of the major scale. Therefore, all scale choices within the major scale’s key should work pretty well. Let’s forget all but the V, or the mixolydian. A C mixolydian should work over the Emi7b5. (If that is confusing, just take a minute and think about it). Taken a step further, the C bebop dominant should be extremely effective over the Emi7b5 chord. With that in mind, go back to all the Bebop licks you worked out from my past lesson (you DID practice, didn’t you?), and use them over the corresponding mi7b5 chords!
If you’re stuck, here’s a way to think about it: Over a mi7b5 chord, use a Bebop Dominant scale whose root is a MAJOR 3RD below. Try it…and please let me know how it works!