No component greater than 1 in absolute value

One of the singular values is 0 if and only if one column is a linear combination of the other two columns.

 [-1+1j    -1j  -1+1j; -1+1j   1-1j  -1+1j;   -1j   1+1j    -1j;  1-1j    -1j   1-1j]     4 2 0
 [ 1+1j   1      1   ; -1     -1+1j  -1+1j; -1+1j   1-1j   1-1j; -1      1+1j   1+1j]     4 2 0
 [ 1-1j  -1-1j  -1   ;    1j   1      1-1j; -1+1j   1+1j   1   ;  1-1j  -1-1j   1+1j]     4 2 0
 [-1-1j    -1j   1+1j; -1+1j  -1-1j   1-1j;  1-1j  -1     -1+1j;  1      1+1j  -1   ]     4 2 0
 [ 1-1j  -1+1j  -1+1j; -1     -1-1j    -1j; -1-1j   1+1j   1+1j;    1j  -1+1j  -1   ]     4 2 0
 [   1j   1-1j   1+1j;   -1j   1+1j  -1+1j; -1+1j    -1j   1   ;  1+1j   1-1j   1+1j]     4 2 0
 [ 1-1j   1+1j    -1j;  1-1j   1+1j    -1j;  1+1j  -1+1j  -1-1j;    1j  -1      1-1j]     4 2 0
 [-1-1j   1       -1j;  1+1j  -1-1j  -1+1j;   -1j  -1+1j   1+1j; -1     -1-1j  -1+1j]     4 2 0
 [-1-1j  -1+1j   1-1j;   -1j   1+1j    -1j;  1-1j  -1-1j   1+1j; -1      1-1j  -1   ]     4 2 0
 [-1-1j  -1      1   ;  1+1j   1     -1   ; -1-1j  -1+1j   1-1j; -1-1j   1+1j  -1-1j]     4 2 0
 [ 1-1j     1j  -1-1j; -1-1j  -1     -1+1j; -1-1j  -1+1j  -1+1j;    1j   1-1j   1   ]     4 2 0
 [ 1+1j   1-1j   1-1j;  1+1j   1      1-1j;  1-1j    -1j  -1-1j; -1      1-1j     1j]     4 2 0
 [ 1-1j  -1     -1+1j;  1+1j  -1     -1-1j;  1-1j  -1+1j  -1+1j;  1      1-1j  -1   ]     4 2 0
 [ 1+1j   1      1-1j; -1+1j   1      1+1j; -1      1-1j     1j;  1-1j  -1+1j  -1-1j]     4 2 0
 [-1+1j  -1-1j  -1+1j; -1-1j   1     -1-1j;  1      1+1j   1   ; -1-1j  -1     -1-1j]     4 2 0
 [-1-1j  -1-1j  -1+1j; -1     -1-1j  -1+1j; -1-1j   1       -1j;   -1j   1+1j   1-1j]     4 2 0
 [-1      1+1j  -1-1j;  1-1j     1j    -1j;  1      1+1j  -1-1j; -1+1j   1-1j  -1+1j]     4 2 0

For these, the three columns are linearly independent.

 [   1j  -1-1j  -1+1j; -1-1j   1     -1-1j; -1-1j   1-1j  -1   ;   -1j     1j    -1j]     4 1 1
 [ 1       -1j   1   ;  1-1j  -1+1j  -1+1j; -1-1j    -1j  -1+1j;   -1j  -1       -1j]     3 2 2
 [ 1-1j   1+1j   1+1j; -1+1j  -1+1j  -1-1j;  1-1j   1+1j   1-1j; -1-1j   1+1j   1+1j]     4 2 2
 [-1+1j    -1j  -1   ; -1+1j  -1-1j  -1+1j;    1j  -1+1j     1j;   -1j   1     -1-1j]     4 1 1
 [-1      1+1j  -1-1j;  1-1j  -1+1j   1+1j;    1j   1+1j  -1-1j;   -1j   1+1j  -1-1j]     4 2 1
 [-1+1j   1+1j  -1   ; -1+1j   1+1j    -1j; -1+1j   1+1j   1   ;  1-1j   1-1j  -1+1j]     4 2 1
 [  -1j  -1+1j  -1+1j; -1+1j   1+1j    -1j; -1-1j  -1-1j   1-1j;  1+1j  -1+1j   1+1j]     3 3 2
 [ 1+1j  -1-1j  -1-1j;  1+1j   1+1j   1-1j; -1+1j   1+1j   1-1j; -1+1j   1+1j  -1-1j]     4 2 2
 [ 1     -1-1j   1+1j; -1      1+1j  -1-1j; -1+1j   1-1j  -1-1j;  1      1+1j  -1-1j]     4 2 1
 [ 1-1j     1j   1+1j;   -1j     1j  -1   ;  1      1       -1j; -1+1j   1+1j  -1-1j]     3 2 2
 [ 1-1j   1+1j   1-1j;  1+1j  -1-1j   1   ; -1+1j     1j   1-1j;  1-1j   1-1j   1+1j]     3 3 2
 [ 1-1j  -1+1j  -1+1j; -1+1j  -1+1j   1   ;  1+1j  -1+1j   1+1j;   -1j  -1+1j  -1-1j]     3 3 2
 [ 1-1j   1-1j  -1+1j; -1-1j  -1+1j   1-1j;  1-1j  -1-1j   1-1j;  1-1j  -1+1j   1+1j]     4 2 2
 [ 1      1-1j   1+1j;    1j  -1-1j   1-1j;  1     -1+1j  -1-1j; -1+1j   1-1j   1-1j]     4 2 1
 [ 1+1j    -1j  -1-1j;  1-1j  -1+1j   1-1j; -1     -1+1j  -1-1j;  1+1j   1+1j   1-1j]     3 3 2
 [-1-1j   1-1j  -1-1j; -1+1j  -1+1j  -1-1j;   -1j  -1+1j  -1+1j; -1-1j   1     -1+1j]     3 3 2
 [  -1j   1+1j  -1   ;  1+1j  -1-1j   1+1j;  1     -1      1-1j; -1+1j   1        1j]     4 1 1
 [   1j    -1j   1-1j; -1-1j   1+1j   1+1j;   -1j  -1+1j     1j;  1-1j     1j     1j]     4 1 1
 [ 1-1j   1-1j   1+1j;  1     -1-1j   1+1j;   -1j   1-1j  -1+1j; -1     -1+1j   1-1j]     4 2 1
 [ 1+1j  -1      1+1j;  1-1j   1-1j     1j;  1-1j  -1-1j  -1+1j;  1-1j  -1+1j  -1-1j]     3 3 2
 [   1j  -1-1j  -1+1j;  1-1j   1-1j  -1-1j;    1j    -1j   1   ; -1     -1       -1j]     3 2 2
 [ 1-1j     1j   1+1j; -1-1j  -1-1j  -1-1j;  1+1j  -1-1j   1-1j;  1-1j  -1-1j     1j]     3 3 2
 [ 1+1j   1+1j   1   ; -1+1j   1-1j  -1+1j;    1j  -1-1j   1-1j; -1-1j  -1+1j   1+1j]     3 3 2
 [-1+1j     1j  -1-1j; -1-1j  -1+1j  -1-1j; -1+1j     1j  -1-1j; -1-1j   1      1-1j]     4 2 1
 [-1+1j   1-1j  -1+1j; -1-1j  -1+1j  -1+1j;  1+1j  -1+1j  -1-1j; -1+1j   1-1j   1+1j]     4 2 2
 [-1-1j   1+1j  -1+1j;  1+1j   1-1j  -1-1j;  1+1j  -1      1+1j; -1+1j  -1+1j    -1j]     3 3 2
 [ 1-1j  -1-1j   1-1j;  1+1j  -1-1j  -1   ; -1+1j  -1-1j   1+1j;  1-1j    -1j  -1+1j]     3 3 2
 [-1+1j  -1     -1+1j;  1+1j   1+1j     1j;  1      1      1   ;   -1j  -1-1j  -1-1j]     4 1 1
 [-1-1j  -1-1j  -1-1j;   -1j    -1j   1-1j;  1+1j   1      1   ; -1     -1-1j  -1   ]     4 1 1
 [-1-1j  -1-1j  -1+1j; -1     -1+1j   1-1j; -1      1-1j  -1+1j; -1     -1-1j   1+1j]     4 2 1
 [   1j   1     -1   ;  1+1j   1+1j   1-1j;  1        1j     1j; -1+1j   1      1+1j]     3 2 2
 [-1      1-1j  -1+1j;  1-1j  -1-1j  -1+1j; -1+1j  -1+1j  -1-1j; -1+1j  -1-1j  -1   ]     3 3 2
 [ 1     -1-1j  -1-1j;    1j   1+1j   1+1j;  1-1j  -1-1j   1-1j;  1      1+1j   1+1j]     4 2 1

No component greater than 2 in absolute value

Only the first example has a zero singular value, and you will note that $1 + j$ times the first column plus $j$ times the second column equals the third column.

 [ 1-2j   1+2j   1   ;  2+1j  -2+2j  -1+1j; -1+2j   1-2j  -1+2j;  1+1j  -1        1j]     6 3 0

For the balance of these examples, the three columns are linearly independent.

 [ 1-2j  -2-1j  -1   ;  2      2+2j   2   ; -2-1j  -1     -2-1j;  1-1j    -1j  -2-2j]     6 3 2
 [ 2-1j     2j  -2   ;  2-1j  -1-1j   2+2j; -2+1j     1j   1+1j; -2+1j  -2-2j     1j]     5 4 3
 [ 2-1j  -2+1j  -1   ; -2+1j   1-2j   2-1j; -1      2-2j   2+1j;    2j   1-2j   2-1j]     7 2 1
 [ 2+1j  -2-1j  -2-1j; -1      1-2j  -2+1j; -2+1j  -1     -2+2j; -2     -2-1j    -1j]     5 4 3
 [-1+2j   2-2j    -1j;   -1j     1j   2   ;   -2j  -2-1j   1-2j;  1-1j     2j   1+1j]     5 4 1
 [ 1+1j    -2j   1+2j;  1+1j   2+2j   2-1j;    2j  -2     -1-2j;  1+2j   1+1j   2   ]     5 4 3
 [-1     -1-2j  -2-1j;  1+1j  -1+2j   2+1j;   -2j    -1j   1-2j;  2-2j     2j     2j]     6 3 2
 [-1      2-1j   2+2j; -1      2+1j   2-1j;  2+1j  -1+2j   2+1j;    2j     2j   2+2j]     6 4 2
 [ 1+2j   1        1j;   -2j     2j   1   ;  2+1j   2-2j   1-1j;  1-1j   1      2-2j]     5 4 1
 [ 1     -2+2j     2j;  1-2j  -2+1j   2   ;  1+2j   1+1j   2   ;    2j   1+1j    -1j]     5 4 2
 [ 2+2j    -1j     2j;   -1j    -1j   2-1j;    1j     1j    -2j; -1-1j   2-1j   2-1j]     5 3 2
 [   2j    -2j  -2-1j;   -2j  -1+1j  -2-1j; -2+2j  -1+1j    -1j; -1-1j   2-1j  -2-2j]     5 4 3
 [ 1-1j  -1+2j  -2+1j; -1+1j  -1+1j     1j;  1+2j     2j     2j; -1+1j  -1     -1-1j]     5 3 1
 [ 1+1j   2+2j   1-1j;  1-2j     2j     2j;  2-1j   2-1j  -1+1j; -1-2j  -1      1-1j]     5 4 2
 [-1-2j   1-2j   1+1j; -1-2j    -1j   1+1j;    1j  -1-1j     2j;  2+2j  -1-1j    -1j]     5 3 2
 [-1-1j    -1j  -1-1j;    2j  -2     -1-1j; -1+2j     2j     2j; -1+2j     2j   2-2j]     5 4 2
 [ 2+1j     1j  -2+1j;   -2j   2-1j   2   ; -1-2j    -2j   2-2j;  2-2j   2      1   ]     7 2 1
 [-1+2j   1-2j  -1-1j; -1      1+1j    -2j;  1      1-2j  -1+1j;  1-1j    -1j  -2-2j]     5 3 2
 [   1j  -1+2j  -1+2j; -2+1j  -2+1j   1   ;  1+2j  -1     -2-2j; -2+1j   2-2j  -1   ]     5 4 3
 [ 1+2j  -1     -2   ; -1+1j    -1j     1j;    1j  -2-1j  -2+1j;   -2j  -1-2j    -2j]     5 3 2
 [ 2-2j   2+2j   2+2j;  2-1j  -1+2j  -1+1j;  2      2       -2j; -1-1j  -1+1j  -2   ]     6 4 2
 [ 1-1j     2j  -1+2j;   -2j  -2-1j    -1j;  2-2j  -1+2j   1-1j;  2-2j    -1j  -1+2j]     5 4 3
 [ 1-2j   2+2j  -2+1j;   -1j  -2     -2+1j; -1+1j  -2      1+2j;  2-1j  -1-1j   2   ]     5 4 3
 [ 2-1j  -2-2j     1j;  1+1j     2j  -1+2j;    2j  -2-1j   2-1j;   -1j  -2+1j  -1+2j]     5 4 3
 [ 2-2j   2+2j   2-2j; -1+1j   2-1j   1-2j;    2j  -2      2   ; -2      1+1j  -1+1j]     6 4 2
 [ 1-2j  -1+2j    -2j;  2+2j     1j  -2   ;  1+1j   2+2j  -1+1j;    2j  -2+1j   2-2j]     6 4 2
 [ 2-1j   1      2   ; -2-2j   1+1j  -1-2j;    1j   1-2j  -1-1j;    2j  -2      2+2j]     6 3 2
 [  -1j   2+1j  -1   ;  2-1j  -1        1j;    1j   1-1j   2   ;  2+2j    -1j   2+2j]     5 3 2
 [   2j   1+2j   1-1j; -1+2j   1+1j  -2   ;  1+1j  -1+1j   1+2j;  1      1+2j   1-2j]     5 4 1
 [ 1     -1-2j  -2-1j; -2-1j   1+2j  -1-1j;    2j  -1     -1+2j; -2-2j   2-1j     2j]     5 4 3
 [ 1+1j   1-1j   1+1j;    1j   2-1j  -2+2j; -1-2j   2-2j  -1-2j;  1+1j     1j     1j]     5 4 1
 [ 2-2j   2-1j   1-2j;   -2j    -1j   2+2j;  1+1j    -1j    -2j; -1-2j  -1+2j  -1-1j]     5 4 3
 [-2     -2+2j     2j;  2-1j   2      2+2j; -1+2j  -1+2j   1-2j; -2+1j    -2j  -2-2j]     6 5 2
 [ 2-1j   2-1j   2   ;    2j   2+1j  -1-2j;   -2j  -1-1j  -2-2j;  1     -1-2j   2-2j]     6 4 2
 [-1      2+2j  -1+2j; -1-1j  -2+1j   2+1j;  2-2j   2+1j  -1+1j; -2        1j  -2   ]     5 4 3
 [-1        2j   1+1j;    1j   2-1j  -2+2j;  2     -1-2j   1+1j;  1+2j   1-2j  -2   ]     6 3 1
 [ 2+1j    -2j  -1+2j; -1+2j  -2-1j  -1-2j; -2     -1-2j  -2+1j;   -1j     1j  -2+1j]     5 4 3
 [-2-1j   2     -1-1j; -2-1j  -2+1j    -2j;   -1j  -2-1j   2+2j;   -2j  -2      1+1j]     6 3 2
 [  -2j   1+2j  -1-2j;  2-1j     2j  -1-2j; -1+2j   1-1j   2+1j;  2+1j  -2      2-1j]     7 2 1
 [-2-2j  -1+1j   1   ; -1-1j   2     -1-2j; -1-1j   2+2j  -2+1j; -1+2j     2j     2j]     5 4 3
 [   2j  -2        1j;  2        2j  -1+1j;  1-1j  -1-2j   2+2j; -2      2+2j     2j]     5 4 3
 [ 1     -1+2j   1-2j; -1-1j    -1j   1+1j; -2+1j  -1-2j  -2   ;  1-1j   2+1j   1+2j]     5 4 1
 [-1+1j  -2+2j  -2+1j; -1-1j  -1-1j   1+1j; -2      2-1j   1+1j; -2-2j     1j  -2   ]     5 4 2
 [-1+2j   1+2j     1j;  1     -2+1j  -1-1j; -2+1j     1j    -1j; -1+1j   1+2j  -1-2j]     5 3 2
 [ 1+1j     1j     2j;  1      2+2j  -2+2j;  1-2j  -2+1j    -1j; -1-1j  -2-2j  -1+2j]     5 4 3
 [-2      1+1j   2+2j;  1-2j  -1-1j   1+2j; -1-1j  -1+2j   1+2j; -2-2j   1-2j   2-1j]     6 4 2
 [ 1+1j  -1      1+1j;   -2j  -1-2j  -1-1j;    2j    -2j   1-2j; -1+2j   2+2j  -2-2j]     5 4 3
 [-2+2j   2+2j   1+2j; -1+1j  -1-1j   2   ;  2-2j     1j  -1-2j;   -2j   2      1+2j]     6 4 2
 [   2j   2+1j  -2   ;  2-2j  -1+2j  -2-2j; -1-2j    -1j  -1+2j;  1     -2-1j  -1-2j]     6 4 2

No component greater than 3 in absolute value

As the components increase in magnitude, it becomes more and more difficult to find such examples, so here are four:

 [-3+2j     3j     3j;    3j  -1     -3-3j;  1+3j  -3+2j   3   ;    2j  -1     -2+1j]     7 6 4
 [-1-3j   3     -2+3j; -3     -3-1j  -1+1j; -1+3j  -1-2j  -3-2j; -3      3+3j   2+2j]     8 6 4
 [-3+1j  -2+1j  -1+1j; -1+3j  -2       -1j;  1+2j     3j   3-2j; -2+2j  -3     -2+2j]     8 4 2
 [-2      1      3-2j;  1+3j   2-3j     1j; -3     -1+3j   2-1j; -2+3j   3     -3-2j]     9 4 2