It seems to be true that multiply transitive permutation groups have been classified completely (using CFSG), but I am having trouble finding a reference where this classification is actually stated. Is there a canonical reference?

2$\begingroup$ @DerekHolt If by "multiply transitive" you mean "2transitive", then you don't want anything else. I see no list of 3transitive groups there. $\endgroup$ – Igor Rivin Nov 4 '15 at 13:33

2$\begingroup$ The descriptions tell you which are 3transitive! These are $A_n,S_n$, some affine groups of eevn degree, some subgroups of $P{\Gamma}L(2,q)$ (I guess you have to work out which), the Mathieu groups. $\endgroup$ – Derek Holt Nov 4 '15 at 13:37

3$\begingroup$ @GeoffRobinson I agree. The thing is that even the list in D&M is very discoursive. If I just want to put in a reference in a paper saying: these are the 2 3 4 5 transitive groups (see \cite{greatReference}), someone who actually cares would have to spent a fair bit of time trying to sort things out in D&M. It would be nice just to have a Landaustyle telegraphic statement. $\endgroup$ – Igor Rivin Nov 4 '15 at 13:57

5$\begingroup$ @DerekHolt And "I guess you have to work out which" speaks exactly to my point. $\endgroup$ – Igor Rivin Nov 4 '15 at 13:57

3$\begingroup$ math.stackexchange.com/questions/698327 seems to give an accurate description of the finite $3$transitive groups, although $A_n$ and $S_n$ are $3$transitive only when $n \ge 5$ and $n \ge 3$, respectively. I still don't think you can improve on D&M for the 2transitive groups, and 4 and 5transitive lists are easy. I admit that 3transitive is a little trickier to get right, mainly because of the complications with $P{\Gamma}L(2,q)$. $\endgroup$ – Derek Holt Nov 4 '15 at 15:13
Here is a list of the finite $3$transitive groups, derived by looking through the list of $2$transitive groups in Section 7.7 of Dixon and Mortimer and identifying those that are $3$transitive.
Let's first recall the structure of $G := P{\Gamma}L(2,q)$ with $q=p^e$, $p$ prime. Let $S={\rm PSL}(2,q)$. For $q$ even, $G = S \rtimes \langle \phi \rangle$ with $\phi$ acting as field automorphism of order $e$, and $G/S \cong C_e$.
For $q$ odd, $G = S\langle \delta,\phi \rangle$, where $\delta$ acts as a diagonal automorphism of order $2$, and $\phi$ as a field automorphism, (Note that this extension is nonsplit when $e$ is even.) We have $G/S \cong C_2 \times C_e$. The subgroup $ S \langle \phi \rangle$ of index $2$ in $G$ is denoted by $P{\Sigma}L(2,q)$.
So now, the finite $3$transitive groups are as follows.
$A_n$ ($n \ge 5$), $S_n$ ($n \ge 3$), degree $n$. (There are two inequivalent actions, conjugate under an outer automorphism of $S_6$, when $n=6$.)
$A{\Gamma}L(n,2) = {\mathbb F}_2^n \rtimes {\rm GL}(n,2)$ with $n \ge 2$. degree $2^n$.
${\mathbb F}_2^4 \rtimes A_7$, degree $16$.
Groups $G$ with ${\rm PSL}(2,2^e) \le G \le P{\Gamma}L(2,2^e)$, degree $2^n+1$.
For $q$ odd, groups $G$ with ${\rm PSL}(2,q) \le G \le P{\Gamma}L(2,q)$ and $G \not\le P{\Sigma}L(2,q)$, degree $q+1$.
The Mathieu groups $M_{11},M_{12},M_{22},M_{22}.2 = {\rm Aut}(M_{22}), M_{23},M_{24}$, degrees $11,12,22,22,23,24$.
$M_{11}$, degree $12$.
For completeness, the finite $4$transitive groups are: $A_n$ ($n \ge 6$), $S_n$ ($n \ge 4$), $M_{11},M_{12},M_{23},M_{24}$,.
The $5$transitive groups are: $A_n$ ($n \ge 7$), $S_n$ ($n \ge 5$), $M_{12},M_{24}$.
And the finite $k$transitive groups for $k \ge 6$ are: $A_n$ ($n \ge k+2$), $S_n$ ($n \ge k$).

$\begingroup$ The precise list should mention the pair (group, set with a 3transitive action) and not only the group. Is it true that in all cases, the stabilizer is unique up to conjugation? if not, up to automorphism? $\endgroup$ – YCor Nov 5 '15 at 10:28

2$\begingroup$ @YCor There is a unique conjugacy class of stabilizers in all cases except for $A_6$ and $S_6$ in degree $6$, in which case there are two such classes fused under an auromorphism. I have added a note about that. $\endgroup$ – Derek Holt Nov 5 '15 at 10:38