How do I solve for displacements after getting my stiffness matrix [K]?

If you end up with a singular stiffness matrix then you should carefully check the boundary conditions. The system is likely underconstrained.

So, in other words, the above methods I used should work if I have my stiffness matrix properly constrained?

Right now I'm running the program on a single Q8 element. Each node has two in plane translational Dof's. I've constrained the bottom three nodes in both directions, and applied a load on the top right corner of the Q8, acting laterally. I feel like the model would be constrained, could you offer any insight into how to constrain it properly?

Yes, the stiffness matrix will become non-singular if you properly apply boundary conditions. It could be singular if you were solving buckling or natural frequency problem but that's not the case here.

Was this element's formulation taken from some Matlab library or defined from scratch by you ?

There are two books that can be particularly useful for you:
- "Introduction to Finite Element Analysis Using Matlab and Abaqus" A. Khennane
- "The Finite Element Method Using Matlab" Y.W. Kwon, H. Bang

They discuss plane stress/strain elements as well (especially the first one may help).

It must be the way I defined the element. I used a more complicated mesh from an actual meshing algorithm and got a non-singular matrix that yielded reasonable displacement values.

Thanks for recomending those books, had one of them on my wishlist already.