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This looks like a bug to me.

The problem is that SageMath doesn't understand the return type. You can work around it as follows:

sage: from sage.interfaces.singular import SingularElement
sage: SingularElement(singular, 'list', 'invariant_ring(%s)' % g.name(), False).sage()
[[            x0 x0^2*x1 - x1^3], [1]]

The problem has been reported as trac ticket #28386.