Exel's notion of associativity (for what it's worth, I know that paper quite well) is just a disjunctive combination of "my" notions of strong associativity and left/right dissociativity, and you're right that I've not included the last two in the OP (basically, for the reason that I mention in the first comment to Andreas Blass' answer). On another hand, I don't agree that sgrpds, as they're defined in Exel's work, generalize sgrps in the same way as grpds do with grps, but this may depend on our relative definitions: For me, a grpd is a cat all of whose arrows are iso. I'm editing the OP.

[...] the notion of subobject that is "naturally" implied by looking at the question from the "privileged" perspective of categories. Do you/anybody know if this has been "corrected" in the book?

@Boris. Thanks, but I may have problems in finding a copy of the book that you mention as a last entry in your list. Yet, I see from the 1988 English translation of Evseev's survey that he's using the term groupoid in the sense of Bourbaki's magma. That's fine, but I find the definition of a subgroupoid provided in the paper unconceivable from any conceptual point of view: Evseev lets a submagma of a magma $\mathbb M=(M,\star)$ be a subset $N$ of $M$ s.t. $a\star b\in N$ for all $a,b\in N$ s.t. $a\star b$ is defined in $\mathbb M$. In my view, this is "wrong", for it doesn't agree with [...]

Yours is precisely my notion of dissociativity, which I've not included in the list in the OP since I myself don't find it particularly "natural". And although we agree that one "right" definition, of course, does not exist, I'd be quite surprised if a "categorial" perspective on the question cannot provide a conceptual motivation, going beyond the scope of a mere "principle of local utility", for favoring one choice over another (which is really what I'm looking for). In any case, thank you much for your answer and the reference.

Many thanks for the references. I will look at the papers tomorrow, but I don't think that the authors give any conceptual motivation to support their own definitions, right?

@Yemon. Yes, but this doesn't count for more than an analogy, as far as I'm concerned. That is, it is not enough to me to justify the use of such terminology. I would use it only if something in the lines of what I wrote above may be true. But this doesn't seem to be the case.

I'm not very happy with the term singular for it would tempt me to refer to non-singular arrows as regular, which is already of common use in algebraic geometry. Moreover, considerations similar to those in the OP can be repeated for the class $\mathcal P$ of all $\bf C$-morphism that are neither left nor right cancellative. So then, how to call the latter if the ones from $\mathcal S$ are named singular? I'm editing the OP and add the question to the rest.

Yes, of course, but we will agree that, though worth mentioning, this is not really different, in some sense, from the example of compact operators, all the more that it can be further abstracted by referring, say, to the ideals of an arbitrary cat.

@Martin. I've edited the title and added some more comments to clarify my point.

@Yemon. No need for feeling embarrassed: We're men, and have all the right to make honest mistakes. Btw, here is a possibly simpler construction: The linear operator $f:\ell^2(\mathbb R)\to \ell^2(\mathbb R): (x_n)_{n=1}^\infty \mapsto (x_n/n)_{n=1}^\infty$ is compact, injective and has dense range.

@Yemon. I'm probably missing something big here, but it is known, e.g., that for any separable Banach space $\mathcal X$ (over the real/complex field) there exists a compact (linear) operator $f: \mathcal X \to \mathcal X$ that is injective and has dense range; see ams.org/journals/proc/2006-134-05/S0002-9939-05-08084-6/… (Proposition 2.1).

@Martin. But nobody (apart from you) claimed that the question of characterizing monos and epis in the semicat of normed spaces and compact operators is "difficult" (whatever this may mean, and for sure it means nothing absolute or definite), and the question raised by the OP is different. So what is the point with your comment?

Btw, it is rather clear that at least all compact operators with dense range must be included, for there exists an obvious faithful (semi)functor from the semicat of real/complex normed spaces and compact operators to the usual cat of real/complex normed spaces and bounded (linear) operators (where epimorphisms are known to be all and the only bounded operators with dense range), and faithful (semi)functors reflect both monic and epic arrows (just like in the categorial case).