I took 16 chord types and found all 62 ways to make them with at least
1 open string and within the first 3 frets.  These are the chords
which will generally form barre chords the best.  E.g. since C7 is
0001, you can form D7 by adding 2 to everything to get 2223.  This
means you play fret 2 on the G-string, 2 on the C-string, 2 on the
E-string, and 3 on the A-string.  Use your index finger to cover all
the 2s then use your middle finger to add the 3 on the A-string.

Chord inversions are ignored in this list.  I.e. in the 0100 voicing
of A7 the lowest-pitch note is C# not A so this might more properly be
called A7-first-inversion (or A7/C#).

The code generating these is at
http://github.com/dancor/uke-chords

major (6+1)
0003: CM
2220: DM
0331: EbM
2010: FM
2013: FM
0232: GM
2100: AM

minor (6+1)
0333: Cm
2210: Dm
1013: Fm
2120: F#m
0231: Gm
2000: Am
2003: Am

diminished (4+1)
2320: Ebdim
2020: F#dim
2023: F#dim
0131: Gdim
3101: Bbdim

augmented (3)
1003: Caug
2110: C#aug
0332: Ebaug

dominant seventh (4)
0001: C7
1202: E7
0212: G7
0100: A7

major seventh (5)
0002: CM7
1302: EM7
0222: GM7
1100: AM7
3210: BbM7

minor seventh (4)
1102: C#m7
0202: Em7
0211: Gm7
0000: Am7

minor major seventh (5)
1103: C#mM7
0302: EmM7
0221: GmM7
1000: AmM7
3110: BbmM7

diminished seventh (1)
0101: C#dim7

half-diminished seventh (4) (just increment any of dim7)
0102: C#m7-5
0201: Em7-5
0111: Gm7-5
1101: Bbm7-5

augmented seventh (4) (just raise 5th of dom7)
1001: Caug7
1203: Eaug7
0312: Gaug7
0110: Aaug7

augmented major seventh (5) (just raise 5th of M7)
1002: CaugM7
1303: EaugM7
0322: GaugM7
1110: AaugM7
3220: BbaugM7

ninth (3)
3203: C9
0310: F9
1021: Ab9

major ninth (1)
1031: AbM9

minor ninth (1)
2032: Am9

dominant seventh sharp ninth (hendrix chord) (3)
3303: C7+9
1310: F7+9
1022: Ab7+9