Sustain is the attitude of a string to sustain it's vibration.

Sustain mainly depends on how much of the string's energy is absorbed and not returned by the material the instrument is made of.

The law which rules the relationship between a material and tone may sound like this: the higher the mass, the higher the retention of energy, the higher the sustain of the note.

In other words the denser and stiff the material an instrument body and especially the neck, is made of, the less of the string's vibration will be absorbed by the material itself (instrument) and the more the resonant peak of the wood will raise over the resonant peak of the strings allowing them to vibrate freely with no interferences, which translates into a longer string's vibration (sustain).

If the note is allowed to vibrate freely, we achieve definition and a balanced tone which is the perfect ground to build up our tone.

Sustain can be reduced or increased by a rather infinite series of factors among which:

The more the thickness of the neck, the more dense and stiff it will be and the longer the sustain will last.

Pickups are magnets. If we place the pickups too close to the strings they will partially limit their vibration. For this reason it is important to find a good compromise between sound and distance of the pickups to the strings especially the neck pickup that influence string's vibration more than the others.

Neck through construction tends to produce more sustain. This type of neck is usually made of many different wood laminations glued together and it is often reinforced with graphite bars as it is not possible to replace it.
All this along with the fact that the vibration travels uninterrupted from the nut to the bridge, raises the resonance peak over the strings resonance thus increasing sustain.

Heavier strings are more tense and deliver more sustain.

Everything else being the same (strings diameter and pitch) the longest the scale the more the tension on the strings the longer the sustain. Please note: the non playing portion of string beyond the nut and the bridge saddles does not have any impact on strings tension, it only affects the "feel" on the fretboard especially bendings (please read here).

Guitar sustain note image
Sustain is the ability of a string to sustain its vibration
(pictured Frudua Carved Slave Pentabass)

Headstock also has an important role on the note sustain. A stiffer neck/headstock structure will help the strings to sustain vibration better. A weak headstock which deflects during playing, will absorb strings vibration and therefore inhibit sustain.

Reinforced headstock on a Tuscany Rush model.

The way the frets are installed into the fretboard slots may have an impact on the neck's stiffness and therefore on sustain.

Too deep frets slots weaken the neck stability causing it to deflect and absorb string's vibration. To avoid this, frets must also be glued into the slots with epoxy resin. The epoxy fills the empty spaces between the frets tangs and the wood preventing the frets from moving and ensuring a good tone transmission.

On the left: a correct depth for fret slot. On the right: too deep frets slots weaken the neck stiffness and reduce sustain.

One of the thing a expert luthier should always look into is how the truss rod is installed into the neck. Neck curvature and truss rod tension are pretty crucial for tone and sustain to avoid unwanted vibrations inside the neck which may act in contrast with some of the string's vibration.

Neck should be build with an up bow relief so that once the strings are at pitch, the correct relief is obtained with the truss rod slightly in tension. The more the truss rod will be in tension with the neck in the correct relief, the stiffer the neck will be, the less vibration of loose truss rod inside the neck and the higher the sustain.

A neck in correct relief with a loose truss rod will deflect more and will most probably show some internal vibration of the truss rod which will inhibit sustain (check sound samples below):
Truss rod loose
Same truss rod in tension.

The neck/body joint also may contribute to some sustain limitations if not properly made. A proper care must be placed in neck's and body wood matching and a particular attention should be applied to the neck's cavity on the body in order to leave it clean from finish particles and to be sure it perfectly fits the necks by adhering on all sides.

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