It might be enlightening to divide equations into further types - first three decidable types, then four undecidable types.

(1) Equations which have a small or otherwise easy-to-find solution

(2) Equations which known techniques prove have no solutions

(3) Equations which (somehow) have an existence proof that a solution exists, but no known explicit solution.

(4) Equations which accepted heuristics suggest should have finitely many or no solutions, and have no small solutions, suggesting there are no solutions at all, but for which no proof that they are unsolvable exists.

(5) Equations which accepted heuristics predict should have infinitely many solutions, but very few of a given size, where we have not found any solutions yet.

(6) Equations which are mysterious in that accepted heuristics predict there should be many solutions, even of reasonable size, but none can be found.

(7) Equations where it is not clear after some thought which heuristics we should believe.

My understanding of what the comments and answers say, in this language, is as follows:

Equations of type (1) and (2) occur at the lowest heights, with (1) occurring at height 0 and (2) occurring at height 1.

We can further subdivide (2) according to the nature of the disproof. According to Bogdan's analysis, the lowest-height unsolvable equations all have mod $p$ obstructions, and next come equations with obstructions arising from divisibility properties of values of quadratic forms (I guess these are probably Brauer-Manin obstructions), and after that equations with obstructions from Vieta jumping. I would guess that at not-much-greater height we will need another obstruction from the toolbox of number theory.

I suspect equations of type (3) exist only for truly absurd heights - one can certainly cook them up using the solution to Hilbert's 10th problem, but I currently can't think of another way.

Somewhere between height 22 and height 45, we get equations of type (4), as I believe Chris Wuthrich's example $x^3-1 -z^2(y^3-1)$ has this form (If we imagine the probability that $z^2 (y^3-1) + 1$ is a perfect cube is proportional to $(z^2 y^3)^{-2/3}$ then summing over $z$, $y$ gives a finite quantity). (Also, Matt F. gave an example of an equation whose nontrivial solutions have this property, but which also has trivial solutions).

Somewhere between height 22 and height 138, we get equations of type (5). The example I'm thinking of is $x^3+y^3+z^3=114$, which is the smallest remaining unknown sum of three cubes. In this case the heuristic of Heath-Brown suggests there should be an (explicit) constant multiple of $\log \log n$ solutions with $x,y,z < n$, so the fact that no solutions are known is consistent with these heuristics.

Equations of type (6) and type (7) surely exist, but could potentially have enormous size.

One reason to study this question further is to understand which of the undecidable types appears first, and which are more common among equations of small height.

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