Or, really, any line of reasoning that seems to ideologically commit us to some sort of ontology that we have no clear-and-substantive relation with? This book from the Cambridge University Press mentions differentiating trivially and nontrivially existent pure numbers (and this arxiv.org essay uses the same phrase, albeit with a perhaps different attached sense). On the other hand, such an option does not appear to be pursued in the SEP article on existence per se, although one might stretch the Meinongian distinction between being and existence in such a manner.
Quine (1948) explicitly characterizes ontology as an attempt to answer the question “what is there?” As he observed, the question may seem deceptively simple, since it may be answered in a word: “everything”. The problem with this answer is that it is largely uninformative. All parties agree that everything is something, but there is still plenty of room of disagreement as to what kinds of objects are there.
Questions for which the answers are absolutely unrestricted are of the kind we elsewise seem to use the word "trivial" to refer to (e.g. trivialism itself is the doctrine that every proposition is true). And so then Quine might be parsed as thinking that existence, as such, is by the by a trivial matter. (He had various issues with modal discourse, so perhaps he really did think modality as a whole is a trivial set of conditions.) I mean, we could rephrase Quine like so:
"What is possible?" "Why, everything."
And it has been said time and again in discussions of omnipotence, for example, that if incoherent descriptions cancel out their intended content, then they do not possibly refer to any integrated descriptum, and so whenever, "Everything," possibly refers, it refers to something itself possible as such: so that possibility as much as, or perhaps even moreso than, actuality is trivialized.
Supposing that a notion of trivial ontology is already current in the literature, does this allow us to move rapidly to a concept of nontrivial ontology, whereby e.g. ontological arguments are undermined insofar as they conflate trivial and nontrivial actuality? I'm not quite sure about this option, appealing as it is, because I'm also unsure how far apart the analytic/synthetic distinction is from the trivial/nontrivial one, and Kant, then, squarely places existential claims on the synthetical, and hence perhaps exactly nontrivial, side of these distinctions.
EDIT: by mentioning trivialism, I did not mean to be arguing for trivialism, but I was using it as an example of how we define triviality in terms of unrestricted options for answers. We often define triviality in terms of things like generic identity, e.g. a trivial elementary embedding is one where j(X) = X, for all X. But then c.f. Frege's universal number: the number of things obeying the law of identity (again, "everything"). My concern with Quine's, "Everything exists," is that this seems to make existence trivial in that, "Some things don't exist," is then always false, and existence is made into something that applies to everything without distinction.
Or then we might think over the difference between the phrase "possible worlds" and the phrase "accessible worlds":
Every world is possible (unless there are impossible worlds, although then we might speak of those as externally possible nevertheless), but not every world is accessible. In fact, the accessibility relation should not be stated "just like that" but always with a qualification like "X is accessible from Y" (by contrast, we could write "X is possible" without reference to any Y).
Similarly, letting ○ and ◊ be actuality and possibility, the inference ◊A → ○◊A is a trivial inference, hence indicates a trivial kind of actuality (I would suppose). Then, against Plantinga's ontological argument, we might say, "It is true that God is actual in all possible worlds, including ours, but this is only insofar as we speak of trivial actuality; substantive actuality is lacking, for God's own world is not accessible from any other possible world" (because divine eternity is not something that creatures are supposed to be able to directly participate in, but there is always to be a gulf between us and It, supposedly).