Use cases for $L^p$ and $l^p$ spaces where $p\neq 1,2,\infty$

Question

Soft question: $L^1,L^2,$ and $L^\infty$ spaces all have many practical uses and an easy intuition behind them (Along with the $l^1$, etc. versions). Just for visualization, I was playing around with $l^1(\mathbb{R}^2),l^2(\mathbb{R}^2), l^p (\mathbb{R}^2), \text{and} l^\infty(\mathbb{R}^2) $ while walking home and considering the distance from (0,0) to (1,1). Trivially we get the general case of $d_p ((0,0),(1,1))=(2)^\frac 1 p$. Intuitionally, I get for $p=1$ we have the taxicab metric, $p=2$ our Euclidean metric, and that for $p>q$, $d_p((0,0),(1,1))<d_q((0,0), (1,1))$ with the limiting behavior being $$\lim_{p\to \infty}d_p((0,0),(1,1))=1=d_\infty ((0,0),(1,1))$$ I can see why we have this monotonically decreasing relationship from the 1 norm to the infinity norm, and the meaning behind the 1, 2, and infinity norms. I was wondering if there is any good intuitions or common use cases for the norms other than $p=1,2,\infty$? They never turned up in my graduate work as concrete things we did anything with, just generic proofs on what would happen if one was bigger than the other.