"What can have happened?" - valid or unnatural?

Solution 1:

(1) Epistemic can

Linguists often distinguish between three types of modality: Dynamic modality is about ability, capacity, physics. Deontic modality is about permission, obligation, social rules. Epistemic modality is about possibility, necessity, knowledge.

The auxiliary can normally carries dynamic and deontic modality.

(1) Mary can swim → ‘Mary has the ability to swim’ – dynamic
(2) Can I leave the table? → ‘Do I have permission to leave the table’ - deontic

However, can does not normally, or at least did not use to, carry epistemic modality.

(3) ?It can rain tomorrow → 'There is a possibility that it rains tomorrow' – epistemic

The meaning of ‘possibility’ for can is not commonly available. Native speakers typically prefer may, might or could in this function instead. Many native speakers will therefore find sentence (3) odd or even unacceptable.

(2) Evidence showing rarity of epistemic can

It can be shown that epistemic can is in fact rare in Present Day English. First, it is far more common to say could have happened than can have happened.

I collected some data from the Corpus of Contemporary American English to demonstrate this:
There is exactly 1 example of can have happened, but 714 instances of could have happened. That’s only 0.1% can have happened.

(5) I can't imagine how that can have happened. (2012, FIC, Analog) (rare)
(6) I don't understand how that could have happened. (2005, SPOK, CNN_Reliable) (common)

The same result is obtained by comparing the results for can have + past participle in general, where there are 130 hits, to could have + past participle, with 41,799 hits. That corresponds to only 0.3% can + perfect have.

Second, there are some studies by professional linguists investigating the semantic distribution of can in large corpora. The following shows the results from one such study:

Fig. 1: The occurrence of can, could, may and might as markers of epistemic, dynamic and deontic modality (from: Collins 2007: 476, table 1).

It shows that can occurs as an epistemic marker in only about 1% of all cases where it is used.

(3) Language change

It has been observed repeatedly over the last decade or so that can seems to take on more and more commonly epistemic functions previously reserved for other auxiliaries. This is very much in accordance with what’s known about the development of modality – dynamic or deontic auxiliaries very frequently acquire epistemic meanings over time (e.g. Traugott 1989). One study finds that the earliest instances of epistemic can may have appeared in the Early Modern English period, rising from 0% in 1420-1500 to 5% in 1640-1710 (Gotti et al. 2002). Coates (1995) claims that epistemic can does not exist, but cites one example of can that could be interpreted epistemically, speculating that such structures might become more common in the future. Her example is the following:

(7) We hope this coding system can be useful [to other linguists working in the field].
(Coates 1995: 63)

A more recent study on conversational data of British English found a “significant increase in the share of can, and a significant decrease in the share of may. [...] This suggests there may be some degree of replacement of may by can, a possibility which is worthy of further investigation” (Bowie et al. 2013).

As things stand, we don’t know exactly how and under which semantic conditions epistemic can may be gaining traction, except that there is good reason to assume that in fact it is.

(4) Speculations on factors promoting epistemic can

I think can may be acquiring epistemic modality through the “non-assertive backdoor”, so to speak, in contexts where possibility is not directly claimed, but indirectly conveyed. Here are the most important cases:

• (a) Negative can has epistemic meaning.

(7) She can’t be home yet. → ‘It’s not possible the she’s at home yet’ - epistemic

• (b) Next, can-questions, especially rhetorical questions, are possible with epistemic interpretation

(8) Can that be true? → ‘Is it possible the this is true’ – epistemic

• (c) In general, interrogatives will be more likely than assertives to develop innovative epistemic can. This is shown not least by the examples brought up in the question. Such questions may not be common yet, but spreading.

(9) a. What can have happened to this woman? → ‘For what things is it possible that they may have happened to this woman’ – epistemic
b. Where can he have gone? ‘For which place is it possible that he may have gone there’ – epistemic

• (d) One might assume that can becomes more frequent in bridge contexts, where both a traditional, dynamic, but also a modern, epistemic reading are available. This could happen in particular with inanimate subjects.

(10) Alcohol can seriously damage your health.
→ ‘Alcohol has the capacity to damage your health’ – dynamic
→ ‘There is a possibility that alcohol damages your health’ – epistemic

• (e) Dummy subjects like existential there should promote epistemic can. That’s because such subjects cannot easily be interpreted as an element that have an ability or permission leaving possibility as the only available interpretation. Such structures do not occur as of yet at a high frequency.

(11) There can be problems in the future. → ‘It is possible that there are problems in the future’ – epistemic

• (f) It may be easier to access epistemic can if it is embedded under words conveying subjective uncertainty, like hope, wish, fear. At the moment, such cases are not common yet (see also Coates’ (7) above).

(12) I fear that Mary can be injured. → ‘(I fear that) it is possible that Mary is injured’ – epistemic

• (g) There may be alternative markers of epistemic modality, such as the adverbs possibly, maybe, whose presence should facilitate epistemic can. Again, such cases are still rare at the moment.

(13) He can possibly be in London. → ‘It is possible that he’s in London’ – epistemic
(14) He can certainly have forgotten about it. → ‘It is possible (to a high degree) that he has forgotten about it’ – epistemic (_{example from John Lawler in the comments})

etc.

Eventually, as epistemic can works its way through these contexts and becomes more and more common, it may also start to occur in unembedded, positive assertives, like It can rain tomorrow, If we continue like this, you can be in trouble, or Mary can be the best student in this class. We’ll just have to wait a few decades and see what happens...

References

Bowie, Jill, Sean Wallis, and Bas Aarts (2013) ‘Contemporary Change in Modal Usage in Spoken British English: Mapping the Impact of “Genre”.’ In: Carretero, Marta, Jorge Arús Hita, Johan van der Auwera and Juana Marín-Arrese (eds.) English Modality: Core, Periphery and Evidentiality. Berlin: De Gruyter, 57-94.

Coates, Jennifer (1995) ‘The Expression of Root and Epistemic Possibility in English.’ In Bybee, Joan and Suzanne Fleischman (eds.) Modality in Grammar and Discourse. Amsterdam: John Benjamins, 55–66.

Collins, Peter (2007) ‘Can / could and may / might in British, American and Australian English: A Corpus-Based Study.’ World Englishes 26.4, 474–491.

Gotti, Maurizio, Marina Dossana, Richard Dury, Roberta Facchinetti and Maria Lima (2002) Variation in Central Modals. Bern: Peter Lang.

Traugott, Elizabeth C. (1989). ‘On the Rise of Epistemic Meanings in English: an Example of Subjectification in Semantic Change.’ Language 65.1, 31–55.

NOTE: I edited this post after it was marked as the “right” answer. I was interested in the topic and so did some more background reading. However, the content has now changed a bit from what was originally accepted.

Solution 2:

Although the title of the question asks if What can have happened? is 'valid or unnatural', at the end of your own question you've reached this conclusion:

one thing I'm sure of is that this construction is by no means unnatural and no less English than the construction with could.

Since you've shown a number of attested examples of the construction in question, I don't think it's debatable whether the construction is valid and natural. It is.

Maybe your question is not whether the construction is valid and natural but how to explain the construction, especially since you're "having a hard time explaining it".

So to better answer your question, I think I should try and critique your own explanation, if I may:

In terms of how I perceive it as a native speaker, all I can say is that it seems to have greater present "validity" than it would with could.

Agreed except for the present part.

When the 'could + have + past participle' construction refers to the past time, it's not could itself but the 'have + past participle' part that is responsible for the past time reference.

By definition, the present perfect is used to refer to an action that took place in the past but has a bearing on the present. As such, I feel it can be questioned from a past or present perspective, depending on the speaker's point of view.

Firstly, the perfect construction following modals is not the present perfect but the "perfect infinitive," although the latter term may be confusing.

Unlike the present perfect, the perfect infinitive is not always "used to refer to an action that took place in the past but has a bearing on the present".

I'm sorry that you can't make the party tomorrow. It would|could|might have been much more fun with you.

I'm sorry that you couldn't make the party yesterday. It would|could|might have been much more fun with you.

So the tense of a modal itself (will|can|may vs. would|could|might) doesn't determine whether the speaker has a present or past perspective.

For instance, if Billy was supposed to be here at 3:00 to unlock the door, I could ask "Where can he have gone?" at 3:00 from the perspective that he IS not here, or "Where could he have gone?" at or after 3:00 from the perspective that he WAS not there at the appointed time.

I disagree. The choice between can and could does not determine whether you have the 'present' or 'past' perspective, because once it's 3 o'clock, you know he's not here, which means that he went somewhere else before the appointed time regardless of when you utter the question.