If my memory doesn't fail me, then to some functional analysts, $L^p$ and $L_p$ spaces are two different things. I understand that many people use $L_p$ to means the space of functions with finite $p$-norm (i.e, $p^\text{th}$-power-integrable functions), while other use the notation $L^p$ for the same purpose. If you are a functional analyst that distinguishes between $L^p$ and $L_p$, then could you please let me know what $L_p$ means (presumably, your definition of $L^p$ coincides with Wikipedia's definition)?
Now, I found another similar notation $\mathscr{L}_p$. What is $\mathscr{L}_p$? There seem to be $\mathscr{L}^p$, $\mathcal{L}^p$, and $\mathcal{L}_p$ too. However, I would expect that this is just due to people's using different fonts, i.e., $L^p=\mathscr{L}^p=\mathcal{L}^p$ and $L_p=\mathscr{L}_p=\mathcal{L}_p$.
The notation "some kind of L" with p "somwhere next to it" is so overused. It can mean:
The space of measurable functions on some measure space with finite $p$ semi-norm. These objects are rarely used on their own. Usually they are a prelimiary steps for constructing Lebesgue spaces.
The Lebesgue spaces. They are a quotient of the former spaces by subspaces of functions with zero $p$ semi-norm. In case of counting measre we use special notation: $\ell_p$.
The Lindenstrauss-Pelczynski spaces. These space on a finite dimensional level level look just like finite dimensional $\ell_p$ spaces. See Absolutely summing operators in Lp-spaces and their applications J. Lindenstrauss; A. Pełczyński, Studia Mathematica (1968)
The non-commutative Lebesgue space constructed for a given von Neumann algebra and a weight. This is a far reaching generalization, where you replace ordinary functions with bounded linear operators. See Non-Commutative Lp-Spaces G. Pisier, Quanhua Xu
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This is not a complete list since there are so many ways to generalize classical Lebesgue spaces. Always check the definition used by your book or article.