Where in the history of linear algebra did we pick up on referring to invertible matrices as 'non-singular'?  In fact, since 


the null space of an invertible matrix has a single vector 
an invertible matrix has a single solution for every possible $b$ in $AX=b$


it's easy to imagine that that invertible matrices would be called 'singular'.  What gives?

If you take an $n\times n$ matrix "at random" (you have to make this very precise, but it can be done sensibly), then it will almost certainly be invertible. That is, the generic case is that of an invertible matrix, the special case is that of a matrix that is not invertible.

For example, a $1\times 1$ matrix (with real coefficients) is invertible if and only if it is not the $0$ matrix; for $2\times 2$ matrices, it is invertible if and only if the two rows do not lie in the same line through the origin; for $3\times 3$, if and only if the three rows do not lie in the same plane through the origin; etc. 

So here, "singular" is not being taken in the sense of "single", but rather in the sense of "special", "not common". See the dictionary definition: it includes "odd", "exceptional", "unusual", "peculiar". 

The noninvertible case is the "special", "uncommon" case for matrices. It is also "singular" in the sense of being the "troublesome" case (you probably know by now that when you are working with matrices, the invertible case is usually the easy one). 
