Turing Machine restrictions that render halting decidable If one restricts Turing Machines to a finite tape (i.e., to use bounded space $S$),
then the halting problem is decidable, essentially because after a number of steps (which can be calculated from the number of states $Q$, and $S$, and the alphabet
size), a configuration must be repeated.


  Are there other natural Turing Machine restrictions that render halting decidable?


Certainly if the state-transition graph has no loops or cycles, halting is decidable. Any others?
A fairly natural and studied variation is the Tape-Reversal Bounded Turing machine (the number of tape-reversals are bounded); see for example:

Juris Hartmanis: Tape-Reversal Bounded Turing Machine Computations. J. Comput. Syst. Sci. 2(2): 117-135 (1968)

Edit: [this variation is more artificial] the halting problem is decidable for a Non-erasing Turing machine that has at most two left instructions on alphabet $\{0,1\}$; see Maurice Margenstern: Nonerasing Turing Machines: A Frontier Between a Decidable Halting Problem and Universality. Theor. Comput. Sci. 129(2): 419-424 (1994)