Which sorting algorithm uses the fewest comparisons?

Merge-insertion sort, a variant of insertion sort, was touted as the sorting algorithm with the fewest known comparisons for several decades, and is perhaps still the best freely-documented sorting algorithm for minimal comparisons:

Merge-insertion sort is the sorting algorithm with the minimum possible comparisons for n items whenever n ≤ 15 or 20 ≤ n ≤ 22, and it has the fewest comparisons known for n ≤ 46.

Glenn K. Manacher's algorithm “and later record-breaking sorting algorithms” (none of which, unfortunately, seem to be freely documented at this time) are reported to adapt the merge-insertion sort algorithm to offer even fewer comparisons.

The winner is Sleep Sort, which uses no comparisons.