$S_4$ has a transitive action of degree $6$, with point-stabiliser given by $\langle (1,2),(3,4)\rangle$. This has no regular subgroup (because the stabiliser does not have a complement, which is easy to see, since the only subgroups of order $6$ in $S_4$ are $S_3$'s, and these all intersect the point-stabiliser non-trivially).
The index $2$ subgroup $A_4$ is also transitive, and so also does not have a regular subgroup. These are the two smallest examples of transitive groups with no regular subgroups.
I just did a computer check, and they also don't have regular subsets.
Here's a short human-checkable proof of this fact in the case of $A_4$:
This action of $A_4$ on $6$ points is imprimitive, with three blocks of size $2$. The double transpositions preserve the blocks, fixing one point-wise and swapping the elements in the other two, the elements of order three permute the blocks.
We are looking for a regular subset $X$, which must have size $6$.
The eight $3$-cycles have only four distinct images for a particular point (since they always map a point outside its block), so there can't be more than four $3$-cycles in $X$.
This means that $X$ contains at least two non-$3$-cycles, but all these have images in common (the double transposition fix some points, so agree with the identity on those, and each pair of double transposition has a block in common which they swap), contradiction.