Let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ such that $a\subseteq b$. This relation defines a lattice structure on $\text{Part}(X)$.


Is there a distributive lattice $L$ such that for no set $X$ there is a surjective lattice homomorphism $s:\text{Part}(X)\to L$?



Quotients of $\text{Part}(X)$
In the paper


Ore, Oystein, 
Theory of equivalence relations,
Duke Math. J. 9, (1942), 573–627


it is proved that $\textrm{Part}(X)$ is simple. (This is very easy to prove.) 


So the only nontrivial quotient of $\textrm{Part}(X)$ is $\textrm{Part}(X)$ itself, which is not distributive when $|X|>2$. Therefore the answer to the question is: any distributive lattice of more than 2 elements is a suitable $L$, and these are the only suitable $L$'s.


