Consider these verbs in past tense:
faxed, emailed, googled
they are all regular verbs made out of new nouns.
Are there any new irregular verbs that I'm not aware of?
I think it's unlikely a new irregular verb will catch on today, given the marked tendency to modify/discard even the ones we still have.
Of the 312 [irregular verbs] which were operative in Old English, only 66 (ie 34%) remain irregular in the twentieth century. And frankly, most of them aren't particularly well-known. Most people don't even realise that wrought, for example, is a past tense form of to work - they just think it's a word that sometimes comes before iron.
You sometimes hear, for example, "thunk" as a "neologistic" past tense for "to think" (similarly snoze, squoze, shat). I doubt such deliberately quirky usage is ever likely to become widespread, but the Internet at large does still seem to be undecided over tweeted, twat, twot, twittered, twitted.
It's worth noting this from one of the best in the field - Steven Pinker...
My example "shat" above may be an exception that proves the rule. Until I read Pinker's summary in the link, I had no real opinion on whether it was ever a "grammatically valid" form. For obvious reasons grammarians of the past might well not have included "to shit" in lists of irregular verbs, even if they knew it was one. Pinker had it as an example of cool, funny, distinctive "neologistically irregular" forms, so I just copied it in. But even he could be mistaken - I know I'm being quirky if I say "The cat shat on the mat", but to my mind the only problem with "Some bloody fox shat on my new decking last night!" is the circumstances causing me to say it, not the grammar itself.
Is it possible for a new irregular verb to appear in English language?
Well believe it or not this question can actually be read multiple ways...
Is it possible for English speakers to begin using a new irregular verb?
Yes! Though it's very uncommon. (In fact it's much more common for irregular verbs to die out and be replaced by regular verbs.)
The most well-known example is regular sneak | sneaked forking into also sneak | snuck beginning in the early 20th century.
Is it possible for a new irregular verb to be considered standard in English?
Possibly, but it ain't easy. Even though snuck has been around for about a century, and many people don't even know the word sneaked, there are still prescriptivists telling people not to use snuck because it's not correct. But despite such complaints, snuck has made it into at least some dictionaries.
Is it possible for a new defective verb to appear in English language?
Defective verbs are verbs which lack one or more standard forms.
Some words or forms entering common usage, especially in the hi-tech realm seem to be somewhere between irregular verbs and defective verbs. They mostly seem to come about when an unhyphenated single-word compound noun related to a hyphenated or two-word compound verb is then verbed back in its single-word form.
I see that's a bit hard to read and understand but let me illustrate:
If log in is the correct base form:
If login is also a correct base form:
So, if login is accepted as correct but in other forms it becomes logs in, logged in, logging in; is it now an irregular verb and/or a defective verb? Good question!!
It is certainly possible for a regular verb to become irregular; it has happened with a few verbs in the United States, most notably sneak/snuck/snuck, but also dive/dove/dived. Some dialects also have drag/drug/drug, this is definitely viewed as non-standard.
Any genuinely new verb will almost have to be regular: that's pretty close to what regular means. But historically, words get taken up, modified, and meanings changed until it isn't entirely clear what the 'regular' form would be. Troubleshoot is an example: is it an offshoot of shoot, with past troubleshot, or a new word, with past tense troubleshooted?
It could happen.
For example, I have observed a lot of confusion/wordplay regarding what the past tense of "tweet" (as in Twitter) should be. I have heard tweeted, twit, and even twat jokingly. I believe twit or twat could (or could have) become standard. Go google "past tense of tweet".
Side note: obviously the issue of difficulty in pronunciation is partly in play for shitted and tweeted. Also, there is some geeky aesthetic in play: cf. plural of Unix and Vax (sometimes Unices and Vaxen)
People enjoy wordplay and with all the neologisms constantly created, it will happen definitively sooner or later if no one can come up with a current example.