1. Littlewood's $4/3$-inequality singles out $\ell^{4/3}$. 

 Namely, given a real valued array $\hat{a}=(\hat a_{m,n}:(m,n)\in\mathbb N^2)$, the norm $\|\hat a\|_{\ell_p}$ is finite for all $\hat a$ such that 
$$\sup \left\{\left|\sum\limits_{m\in\mathcal M,n\in\mathcal N}\hat a_{m,n}x_my_n\right|:x_m,y_n\in[-1,1],\mathcal M,\mathcal N\mbox{ are finite}\right\} < \infty$$
if and only if $p\geq 4/3$.



2. The second example is somewhat tangential to the question but I find it worth mentioning. It is concerned with the peculiar asymptotics of $L^4$-norms of the Hermite functions (see, e.g., [*Lectures on Hermite and Laguerre expansions*][1] by Thangavelu, Lemma 1.5.2). 
>**Proposition.** As $n\to\infty$ the Hermite functions satisfy the estimates 
$$\|h_n\|_{p}\sim\begin{cases} n^{\frac{1}{2p}-\frac{1}{4}}, &  1\leq p< \infty, \\\ \\\ n^{-\frac{1}{8}}\log n, &  p=4, \\\  \\\ n^{-\frac{1}{6p}-\frac{1}{12}}, & 4 < p\leq \infty. \end{cases} $$
Here $a_n\sim b_n$  means $a_n=O(b_n)$ and $b_n=O(a_n)$. 


  [1]: http://books.google.co.uk/books?id=I9Kkb4SnTw0C&printsec=frontcover&dq=lectures+on+hermite+and+laguerre+expansions&source=bl&ots=kL4N2qXmal&sig=ZJPEHMcU4J_HvWaZ74v0q2z95TA&hl=en&ei=BucFTZTQEITr4gbd29i6Cg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CB4Q6AEwAA#v=onepage&q&f=false