One can still quite often read or hear the argument that decoherence solves the measurement problem and therefore further discussion of interpretations is unnecessary.
Fortunately, I can keep this blog post short with a link to this paper:
Stephen Adler, Why Decoherence has not Solved the Measurement Problem.
If one wants to read about the role of decoherence within different popular interpretations, I recommend this paper:
Maximilian Schlosshauer, Decoherence, the measurement problem, and interpretations of quantum mechanics
It notices that "Decoherence adherents have typically been inclined
towards relative-state interpretations ... It may also seem natural to identify the
decohering components of the wave function with different Everett branches." and it then proceeds
to discuss two important open issues of that interpretation: the preferred-basis problem and the problem with probabilities in Everett interpretations.
If one want to go down that route I recommend this paper for further reading.