If the matrix is singular, there are multiple solutions to the set of equations Ax=0.  

As to zero diagonals, most matrix solvers like to have diagonally-dominant matrices, so if they find a zero on the diagonal (may be due to numerical roundoff), then they swap rows or columns such that a large number from another position is placed on the diagonal.

Typically, these sorts of messages appear when you are outside the expected operation of a strongly nonlinear device, and it's not truly a singular matrix, just singular within the finite precision of real numbers on a computer.

If you take a current source and drive the series combination of a 1e-20 resistor and a 1e+20 resistor, then the simulator will be unable to solve the circuit: the (conductance) matrix will be

1e+20	-1e+20
-1e+20	(1e+20+1e-20)

and 1e+20+1e-20 = 1e+20 in double-precision arithmetic, such that the two rows are linearly dependent (row1 = -row2).

If you have a diode with a 0.5 Ohm series resistance, then for certain forward biases the diode conductance gd will be sufficiently large that gd+1/rs = gd (numerically); also, for some reverse biases, gd+1/rs = 1/rs -- though usually the simulator's GMIN prevents this.