This is an active research topic.  I'm currently working on the front line towards a classification of TPMSs of genus 3 (TPMSg3s).  My collaborators include Weber and Traizet.  I also know a Japanese team working on the moduli space.

Recent progress include surprising discoveries of new examples:

* https://arxiv.org/abs/1804.01442 (j/w Weber)
* https://arxiv.org/abs/1807.10631 (j/w Weber)

and rigorous proof of some deformations of the gyroid.

* https://arxiv.org/abs/1901.04006

These examples are important because they reveal concrete singularities in the moduli space of TPMSg3s.  So their existence basically proves that the classification is very complicated.

However, an ultimate classification of TPMSg3s is not completely hopeless.  We are making big progress on the boundary of the moduli space, and we expect to see more new examples in the near future.  I estimate about 5 years of hard works before we can finally evaluate the feasibility of the classification. I'm optimistic.

Joint with Traizet, we also constructed uncountably many examples of non-periodic minimal surfaces.

* https://arxiv.org/abs/1908.06276

Some of them can be regarded as "quasi-periodic", but not "quasi-crystallographic".  I have been thinking about quasi-crystallographic minimal surfaces, but currently have no progress at all.

My involvement in the topic is mostly motivated by physics, and I'm actively collaborating with many experimental teams.