Derivation of least squares solution in case of overdetermined, square, and underdetermined.

Question

Can someone derive (or provide a complete reference for) the least squares solution of a linear system in the cases of overdetermined, square, and underdetermined? I added a link to a solution I compiled that gives the information I want to derive: https://math.stackexchange.com/a/2138532/246090

derive:

1. minimal residual solution family $\hat x \in \hat X$ of overdetermined system Ax=b, when $A^TA$ nonsingular and also derive the solution with minimal $\|\hat x\|$

2. minimal residual solution family $\hat x \in \hat X$ of overdetermined system Ax=b, when $A^TA$ is singular and also derive the solution with minimal $\|\hat x\|$

3. minimal residual solution family $\hat x \in \hat X$ of square system Ax=b, when $A^TA$ is nonsingular and also derive the solution with minimal $\|\hat x\|$

4. minimal residual solution family $\hat x \in \hat X$ of square system Ax=b, when $A^TA$ is singular and also derive the solution with minimal $\|\hat x\|$

5. minimal residual solution family $\hat x \in \hat X$ of underdetermined system Ax=b, when $A^TA$ is nonsingular and also derive the solution with minimal $\|\hat x\|$

6. minimal residual solution family $\hat x \in \hat X$ of underdetermined system Ax=b, when $A^TA$ is singular and also derive the solution with minimal $\|\hat x\|$