Here is a list of CFTs. They are organized by dimension. Note that some CFTs, like the gaussian theories and percolation, can be defined in any number of dimensions.

Some special classes of CFTs:

• Holographic CFTs
• Superconformal field theories
• Logarithmic CFTs
• Topological field theories
• Non-unitary CFTs
• Nonrelativistic CFTs

In 1d, the conformal Ward identity $T_\mu{}^\mu = 0$ implies $T_{00} = H = 0$. Therefore quantum systems which truly respect conformal invariance are non-local (see here for another argument). Nonetheless, there are a number of interesting models exhibiting conformal symmetry or near conformal symmetry in one dimension.

2d CFTs are the best understood class of CFTs, due to their larger Virasoro Algebra. 2d CFTs differ based on their central charge; those with $c<1$ comprise considerably simpler class of Minimal Models.

Those with $c>1$ are more complicated and in general are not fully classified yet.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.31.851

https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.122.211601

https://arxiv.org/abs/hep-th/0608014

https://arxiv.org/abs/1805.06467

There are no known unitary interacting CFTs in $d>6$.

There are constructions of non-unitary theories in higher dimensions (see non-unitary CFTs).

There are no interacting superconformal field theories in $d>6$ because there are no superconformal algebras (satisfying certain assumptions). This follows from the classification of superconformal algebras due to Nahm. See also the proof of Minwalla for a more recent discussion.

These apparently exist, even nonperturbatively. What's the conformal symmetry group? Possibly relevant: http://mr.crossref.org/iPage?doi=10.1070%2FRM1988v043n02ABEH001720

Here's an example of the Ising model on the Sierpinski carpet: https://arxiv.org/abs/cond-mat/9802018.

Here's some bootstrap work: https://arxiv.org/abs/1309.5089.