The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not remember exactly.

I have always been confused by this condition, especially for a sheaf $F$ on a locally Noetherian scheme $X$, mainly because as far as I know, there are at least *three* different definitions in the literature. Let us look at them, $F$ is said to satisfy condition $(S_n)$ if:

$$ depth_x (F_x) \ge \min (\text{dim} \mathcal O_{X,x},n) \ \forall x\in X \ (1) $$

$$ depth_x (F_x) \ge \min (\dim F_x,n) \ \forall x\in X \ (2) $$

$$ depth_x (F_x) \ge \min (\dim F_x,n) \ \forall x\in \text{Supp}(F) \ (3) $$

Definition (1) can be found in Evans-Grifffith book "Syzygies", definition (2) is given in EGA IV (definition 5.7.2) or Bruns-Herzog book "Cohen-Macaulay modules". Definition (3) is what VA used in his answer to the second question quoted above (and I certainly have seen it in papers or books, but can't locate one right now, so references would be greatly appreciated).

When $F$ is the structure sheaf or a vector bundle (of constant positive rank), then they all agree. However, they can differ when $E$ is a sheaf. For example, (1) allows us to say that if $X$ is normal, then $E$ is reflexive if and only if it is $(S_2)$. But according to (2) or (3), if $X=Spec(R)$ for $(R,m)$ local, then $k=R/m$ would satisfy $(S_n)$ for all $n$. (2) and (3) are equivalent if we assume that the depth of the $0$ module is infinity, but I have seen papers which do not use that convention, adding to the confusion.

Since a result surely depends on which definition we use (I have certainly made mistakes because of this confusion, and I think I am not alone), I would like to ask:

**Question**: Is there an agreement on what exactly is condition $(S_n)$ for sheaves? If not, what are the advantages and disadvantage for each of the different definition?

Some precise references:

Bruns-Herzog "Cohen-Macaulay modules" : after Theorem 2.1.15, version (2)

Evans-Griffith "Syzygies": part B of Chapter 0, version (1)

Kollar-Mori "Birational geometry of algebraic varieties": definition 5.2, version (1)

EGA, Chapter 4, definition 5.7.2, version (2).