Usually when you were swearing. And I mean that in both ways.

Getting at everyday, conversational Ancient Greek is perhaps surprisingly tricky. What survives to us is what people wrote, and there's always a disconnect between the language as spoken and the language as written, whatever you're speaking (even in English: when we write, we insist that knight and night are different words, but not when we speak).

Moreover, most of the sorts of writing that we can now read had their own rules of style that would take away the sorts of casual, everyday tics we're interested in here. Epigraphy - public inscriptions - is the obvious one, but this works for poetry, courtroom speeches, scribbled curses, philosophical dialogues and just about everything. This is particularly true of Greek written in the Roman period - which by volume is by far the majority of what has survived - where educated writers were expected to imitate fifth- and fourth-century Athenian Classics, even though their dialect diverged massively from Greek as it was spoken in their own time. There's a good anecdote, which I can't find at the moment, of a competitor at a public-speaking competition (let's call him Diogenes) being pulled up by the judges and asked to name the Classical author in which he found a certain word that he had used in his speech, and responding with 'Diogenes'.

So - this is not going to be perfect. What I'm going to do here is look at some sorts of literature - comedic drama, novels and philosophical dialogue, in particular - that aimed to represent, albeit in their own stylised way, everyday speech.

Something you see regularly in these is the particle ma ('by'). In very Ancient Greek - the Homeric poems, which are largely an artefact of the 8th century BC - we find ma being used when swearing formal oaths. So in the Iliad, Achilles swears ma Apollo to protect the army's prophet from any retribution if he gives an unpopular prediction, and ma the speaker's staff that he is holding to emphasise how serious he is when he withdraws from battle. By the fifth century, we see this used much more casually when you just want to be emphatic. So in one of Aristophanes' comedies (the Thesmophoriazusae), a character is being asked to go to the temple, and refuses with the line 'no, ma Apollo.' You see this sort of use all the time in similar texts, normally with negative statements, when you want to be very clear that you mean 'no'.

However, there seems to have been a trend, particularly in colloquial speech, to drop out the name of the god - after all, you don't want to cheapen a full-blown oath by swearing one over something trivial. So in another of his plays (the Frogs), Aristophanes has his Chorus come out with 'ma... I'd never have believed that, not if someone came up and told it to me'. Similarly, in one of his dialogues (the Gorgias), Plato has Socrates correct someone's inconsistent statement of 'I think...' with 'ma... - no you don't!'.

As for what you might hear people shouting in a fight - the Greek word for 'making a war-cry' is alalazein - that is, literally, 'to go alala' (or alale, depending on which dialect you speak). At least in mythology and heroic poetry, that's what warriors shout when they go on the attack, and it's a pretty safe bet that some real-life warriors would have made life imitate art and copied it. Interestingly, alala was back-engineered into a goddess of war, so by shouting that, you are indirectly shouting the name of a goddess, though it's slightly different to what you've sketched in the question.

In the mythology of the naval Battle of Salamis, won by the Athenians against the Persians in 480 BC, there developed the theme of the Athenians rowing into battle singing a paean, Paeans were originally religious in character and dedicated to Apollo: the lyrics of this one, at least as we have them from the tragic playwright Aeschylus, do mention the gods in general, and the format would be thought to be associated with them:

O sons of the Greeks, go,Liberate your country, liberateYour children, your women, the seats of your fathers' gods,And the tombs of your forebears: now is the struggle for all things.

In other cases, though, chants seem to have been more practical - we often hear of rowers chanting or humming ri-pa-pai to keep in time, and the Spartan poet Tyrtaeus wrote a number of 'inspiring' songs to be chanted by soldiers as they marched, encouraging them but also helping them to keep in step. I don't think any of the bits of those that survive mention the gods, but then most of his poems don't survive at all.

In summary - the gods were everywhere in Classical Greece, and if I'd included in this answer the sort of literature that tries more obviously to elevate itself above everyday speech, you'd find them all over the place. In another answer I wrote on here, I found a fourth-century Greek-speaker in Antioch, who described night-time activities in his city as 'equally split between Aphrodite and Hephaestus' - that was the sort of flowery, poetic flourish that educated, literary people would drop in to their writing, and so presumably their speech, to show off. So while the specific uses in the question might not totally map to what we see in our sources, the general sentiment is pretty sound.

Just to add a little to an already great answer, some patterns do emerge from our literary sources on these kinds of informal oaths (oaths you make not with the full formula of “I swear by whatever god or goddess” but just a simple positive or negative particle and the god). In Oaths and Swearing of Ancient Greece Sommerstein even specifies that informal oaths occur specifically in prose or “one of the less elevated poetic genres” in order to avoid those made in more formal poetic genres.

One of the more interesting things about informal oaths that emerges particularly from comedy and somewhat from our philosophical texts is the high degree of variation in which god or goddess a person swears by. In our earlier comic texts, there is a small group of ‘unisex’ gods sworn by people of both genders, but on the whole women swear by a certain set of goddesses exclusively. Aphrodite, for example, is used in invocation exclusively by women and in one instance in Thesmophoriazousae by a man wearing women’s clothing in order to pretend to be a woman, clearly as a way of gendering his speech. Other goddesses, including Athena and Demeter alone (Demeter and Kore together are only used by women) are only used by men, as are a number of male gods. And it seems this trend only grew stronger with time. By Menander, the only god used in informal oaths by both genders is Zeus, and even then very few instances are by women.

An issue in the Socratic literature also indicates the potential for regional variation that Attic comedy gives us an only partial view into. In Aristophanes’ Lysistrata, Spartan characters of both sexes swear by Castor and Pollux, indicating some attempt by an Attic playwright to introduce regional difference, but the lack of gender differentiation on this oath further reinforces Aristophanes’ lack of familiarity with its use (it’s notable also with Castor and Pollux in particular that in Latin, chiefly appearing in comedy, informal oaths by Castor and Pollux are gendered. pol and edepol, by Pollux, are used by both sexes but ecastor and mecastor, by Castor, are used only by women). On the other hand, in both Xenophon and Plato, there are a notable number of informal oaths made by Hera, a substantial number of them by the character of Socrates. Two of the most recent suggestions for why the character of Socrates in particular should favor oaths by Hera both ascribe it to regional variation. Sommerstein argues that all the characters who use it have a connection to the Attic deme of Alkope - Socrates’ own deme - while Sanders argues on the basis of stronger evidence for the cult’s presence in Sparta than anywhere in Athens, let alone a particular deme, that the oath has a Spartan character.

While the lack of a firm answer to the “by Hera” question highlights some of our difficulties, mostly with absence of evidence, hopefully this gives you a bit of a view into some of the different factors that affected which gods a person swore by when using informal oaths.

If you’re interested in oaths more generally, from swearing them to trying to get out of them, you may want to check out:

Oaths and Swearing in Ancient Greece. Edited by Alan H. Sommerstein and Isabelle C. Torrance. De Gruyter, 2014.