I will start the story by the end:

Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?

The general motivation is that everyone says that they are "mild" singularities, and I would like to know if, for instance, can we deduce that a variety with canonical singularities which is also Gorenstein (and $\mathbb{Q}$-factorial?) is a local complete intersection or not?

The original motivation is that I was looking at some prime nef divisor $D$ in a terminal, $\mathbb{Q}$-factorial, Gorenstein, Fano variety of dimension $n\geq 3$, and I wanted to prove that $D$ it wasn't ample. Finally I arrived to prove that the restriction map $$\operatorname{H}^2(X,\mathbb{R}) \to \operatorname{H}^2(D,\mathbb{R}) $$ it wasn't an injection. But now the problem is that we know that the Lefschetz hyperplane theorem is true if, for instance, $X\backslash D$ is smooth or $X$ is a local complete intersection (see this MO question, for example), which is not my case.

More generally, it has been proven by Goresky and MacPherson in their book "Stratified Morse theory" that if we define a measure $s(p)$ of the degree of singularity of $X$ at a point $p$ to be (the number of equations needed to define $X$ near $p$) minus (the codimension of $X$ in projective space), the we have an analogue

Theorem (the LHT for singular spaces): Let $X$ be a purely $n$-dimensional algebraic subvariety of complex projective space, and let $H$ be a generic hyperplane. Then $H_i(X,X\cap H)=0$ for $i<n-\sup_{p\in X} s(p)$.

So again it arises the question that if we could analyse how far is a variety with "mild" singularities to be a local complete intersection, namely, compute $s(X)=\sup_{p\in X} s(p)$ ?

Thank you very much in advance for your comments and references.