Dominant Bebop Scale
A set of notes commonly played in the bebop style of jazz. Consists of a mixolydian mode with an added major 7th (used as a passing tone between the root and the minor 7th). As with any mixolydian scale, it is used over a dominant 7th chord. It has two distinct useful properties: 1)When played descending, all of the chord tones fall on the beats, enhancing the feeling of tonality. 2) When played ascending, the passing tone (maj 7th) makes the whole scale smoother as you continue up.

C Dominant Bebop Scale
First, we find C Mixolydian, to do this, we use the notes from the F
major scale and apply them to the degrees of the C, in this case: C D E F G A Bb C If we pick out the chord, we get a standard C7, or C E G Bb C. However, to make this a bebop scale, we then add the major
7th of C, which is B. The resulting scale is as follows: C D E F G A Bb B C
Note: this scale can also be played in ANY mode of the original I chord (in this case, the I was F). For example, if the current chord were G-7 (the ii chord of F), I could still play the C bebop scale over it. Note that this scale sounds best when played over the intended V, the ii, the Imaj7, or the vii chords. It is seen most commonly when played over an entire ii-V-I progression (would be based on the V chord).

Major Bebop Scale
Quite similar to the dominant bebop, this scale features a major scale with an additional augmented fifth as a passing tone between the perfect fifth and the major sixth. For example:
C Major Bebop Scale
We find C major, as such:

We then add the augmented fifth (G#):
C D E F G G# A B C
This scale can be played over any major chord: it emphasizes the major sixth chords and uses the augmented fifth as a passing tone.