Q: How can I perform sphereing when the covariance matrix is degenerate?

In this case you can use an SVD. For instance, in matlab do the following:
[U,S,V] = svd(X). This will give you matrices U,S,V such that X=U*S*V'.

Now define Y=sqrt(N)*inv(S)*U'*X, then Y is sphered.

Proof:
Y*Y'/N =
sqrt(N)*inv(S)*U'*X*X'*U*inv(S)*sqrt(N)/N =
inv(S)*U'*U*S*V'*V*S*U"*U*inv(S)=
I !!