The D Shape and the G Shape
The remaining three shapes — D, C and G — are more awkward than E and A, and many players stop after the first two. That is a mistake, but a comprehensible one, so it is worth being honest about what these shapes are good for.
They are rarely played as full six-string barre chords. They are usually played as fragments: three or four strings, higher up the neck, filling a space rather than laying a foundation. That is not a compromise. A three-note chord high on the neck is exactly what you want when a bass player and a keyboard are already covering the bottom.
Start with the D shape. Open D uses only the top four strings.
D
Open D. The root is on the open fourth string, and the sixth and fifth strings stay out of it.
Moved up the neck, it keeps that top-four-string character. Here it is at the fifth fret, where it becomes a G.
G
G major in the D shape, up at the fifth fret. Small, bright, and it sits above anything a bass is doing.
The G shape is the awkward one to move, because open G is spread across the whole neck with a three-fret stretch. As a moveable shape it is usually reduced to its top few strings, which is how it shows up in practice.
G
Open G, the parent shape. Root on the sixth string, third fret.
C
C major in the G shape, as a fragment on the top four strings. The stretch to the highest note is what makes the full version of this shape rare.
A word on what these fragments actually contain. A major chord has three notes — root, third and fifth — and a six-string chord just repeats some of them. Drop a repeat and nothing essential is lost. What you lose is weight, and often that is what you wanted to lose.
So when you see a shape here with two strings muted, do not think of it as an incomplete barre chord. Think of it as the part of the chord that is useful in a band.