Limited number of variable occurrences in 1-in-3 SAT
Up to my knowledge the current "limits" have been settled in:

Stefan Porschen, Tatjana Schmidt, Ewald Speckenmeyer, Andreas Wotzlaw:
XSAT and NAE-SAT of linear CNF classes. Discrete Applied Mathematics 167: 1-14 (2014)

See also Schmidt's Thesis: Computational Complexity of SAT, XSAT and NAE-SAT for linear and mixed Horn CNF formulas

Theorem 29. XSAT remains NP-complete for $k$-$CNF^l_+$ and $k$-$CNF^l$, $k, l \geq 3$.

(XSAT for $3$-$CNF^3$ is exactly 1-in-3-SAT where each variable appears exactly $l=3$ times)

Note that the theorem also proves the NP-completeness of the stronger monotone case ($CNF_+$)