In Oxley's book Matroid Theory (2nd edition) on page 594 there is a table stating that there are 38 non-isomorphic matroids on 5 elements. 
I have found 34: 32 graphic matroids (whose possible associated graphs you find in the pictures above an below).
Two more are the uniform matroids $U_{2,5}$ and $U_{3,5}$.
This leaves 4 more non-isomorphic matroids I would like to see. Can someone point me in the right direction to find these missing 4?
On
Matroids are graphic iff they don't contain any of the five forbidden minors listed here. All but one of those forbidden minors have more than five elements, so the six non-graphic matroids on 5 elements necessarily have $U_{2,4}$ as a minor. So you can easily find those four remaining matroids by starting with $U_{2,4}$ and adding or expanding an element appropriately.
There are 16 binary matroids on four elements, and one non-binary maroid $U_{2,4}$: the uniform matroid on four elements. You can expand $U_{2,4}$ in three standard ways to a matroid on five elements as follows.
Assume ground set $E(U_{2,4}) = \{1,2,3,4\}$. You can add another element $5$ as
This gets you 3 further matroids in addition to the 32 binary ones you have already found. If you take duals of these 3 you get the rest.