There are several approaches to playing jazz on guitar without a rhythm section. The guitar is of course capable of being used as a solo instrument, as shown by the classical repertoire as well as jazz artists such as Joe Pass and Tuck Andress. The technical problem to solve is to how to play chords, bass line, and melody with only ten fingers?
There are two parts to solving this – conceptual and technical, and they are interrelated. A finger-style technique allow the player to keep a bass line going while playing chords and melody, much like classical pieces demand. Using a flatpick, on the other hand, causes the player to have to shift back and forth between playing bass and melody. A third, hybrid method, sometimes called "chicken picking" involves using the pick for bass and several fingers of the plucking hand to play melody and chords.
Conceptually, the choice is this: a consistent bass line while keeping a melody going involves pretty much working out the parts – there isn't much opportunity to improvise while you are doing all that. To have an improvising solo style, you need to switch back and forth – playing enough bass to give the listener the idea of a progression, and enough melody and chords to create good phrases. It's a little bit of smoke and mirrors, but it changes the focus of playing solo from a worked-out-in-advance piece to a looser somewhat on-the-fly version.
The key here is developing compositional ability within improvisation. For instance, using call-and-response is very effective when playing solo. You can state a motif as part of the bass line or on the treble strings, and then answer it in the other register. You can learn to vary the length of phrases, the interplay of single lines and chords, and the interplay of fast passages and simpler passages. However you put it together, solo guitar will challenge you to develop your compositional abilities, especially if you are a guitar player who tends to just 'shred ". Good improvisation is just very fast composing – while you are playing a phrase, part of your brain learns to formulate what you are going to do next, and another part is shaping the solo as a whole. All of this is done intuitively – you don't have time to second-guess or revise. So practicing solo guitar is good practice for improvising in any situation. Remember improvising is not just about learning to run fast scales and arpeggios over chords. It's about creating meaning and melody.
Guitar Right-Hand Styles
There are a lot of choices here and it partly depends on the style of music one plays.
Flatpicking: using a flatpick, and learning to play with up and down strokes to develop speed when playing single lines. Variations include Chickenpicking – using the pick to hit bass notes while the 2nd, 3rd and 4th fingers play single lines. Combination picking – like chicken picking, but the fingers pluck chords and the pick is used when single lines are played. Cross-picking – Playing fast arpeggios with a pick, such as imitating the typical banjo roll on the guitar (a 3-3-2 rhythm) Sweep picking – fast picking of arpeggios using all down strokes or up strokes
Classical style: using the fingers to play both chords and single lines.
Finger picking: Keeping a bass line and melody going, using one of the patterns usually called "Cotton picking" (after Elizabeth Cotton) or "Travis picking" (after Merle Travis, though his style was more complicated than a few patterns). Chet Atkins and Tommy Emanuel are examples of this style. Blues musicians often used a variation on this style in which the thumb did alternate bass note, but just stayed on the root (Lightnin 'Hopkins and Mance Lipscomb, for example).