The sixth spans six diatonic scale degrees, the minor sixth contains 8 semitones, the major sixth contains 9 semitones. The pitch ratio of the minor sixth is 8/5 and of the major sixth is 5/3. 3, 5 and 8 are adjacent numbers of the Fibonacci sequence. So, if we divide the octave by the golden ratio, we'll find ourselves between the minor and the major sixth. Maybe that's why the sixths give a pleasant, harmonic sound.
The major sixth may be found between the 5th and the 3rd of the major triad (G and E in the C-major triad) and between the 3rd and the root of the minor triad (flat E and C in the C-minor triad).
The minor sixth may be found between the 5th and the 3rd of the minor triad (G and flat E in the C-minor triad) and between the 3rd and the root of the major triad (E and C in the C-major triad).
There are countless examples of songs where the sixths are frequently used and that helps to memorize them. My Bonnie begins with a major sixth; G - E.
Mark Knopfler's first solo in Private Investigation starts with a minor sixth; B - G.
The theme in the movie Love Story is full of major and minor sixths; C - E descending, E - C ascending, D - B, C - A, B - #G.
Once you memorized the sixths from the songs, it's easy to hear that jazz and blues musicians often use sixths in their improvisations.
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