Is there an algorithm that finds the forbidden minors?
The answer by Mamadou Moustapha Kanté (who did his PhD under supervision of Bruno Courcelle) to a similar question cites A Note on the Computability of Graph Minor Obstruction Sets for Monadic Second Order Ideals (1997) by B. Courcelle, R. Downey, and M. Fellows for a non-computability result (for MSOL-definable graph classes, i.e. classes defined by a Monadic Second order formula) and The obstructions of a minor-closed set of graphs defined by a context-free grammar (1998) by B. Courcelle and G. Sénizergues for a computability result (for HR-definable graph classes, i.e. classes defined by a Hyperedge Replacement grammar).

The crucial difference between the computable and the non-computable case is that (minor-closed) HR-definable graph classes have bounded treewidth, while (minor-closed) MSOL-definable graph classes need not have bounded treewidth. In fact, if a (minor-closed) MSOL-definable graph class has bounded treewidth, then it is also HR-definable.

The treewidth seems to be really the crucial part for separating the computable from the non-computable cases. Another known result (by M. Fellows and M. Langston) basically says that if a bound for the maximum treewidth (or pathwidth) of the finite set of excluded minors is known, then the (finite) minimal set of excluded minors becomes computable.

It is not even known whether the (finite) minimal set of excluded minors for the union (which is trivially minor-closed) of two minor-closed graph classes each given by their respective finite set of excluded minors can be computed, if no information about treewidth (or pathwidth) is available. Or maybe it has even been proved in the meantime that it is non-computable in general.