Unfortunately, as regards the most common stereotypes, I must challenge the premise of the question; that any of these are 'largely untrue.'.
1. Common stereotype: The Irish are all drunks!
"But Ireland has the highest proportion of non-drinkers per capita!"
Find me a single recent study to support this oft-quoted contention. Please, I'm begging you: I want to shove it down peoples' throats too. But if it was once true, it isn't any longer. Possibly fueling the longevity of this urban legend is the fact that the Catholic alcohol abstinence movement, The Pioneers (not to be confused with various youth organisations in former Communist countries, well, not much at any rate), was formed in Ireland .
Furthermore, anecdotal evidence conducted through the lens of the bottom of my pint glass tells me that even when other, larger countries (e.g. France) drank more than us (in 2000 for example), their pattern of drinking was completely different. To continue with France as an example, French adults would typically drink alcohol every day, but this is a glass of wine with lunch, another with dinner and so on. Irish people may not, as a rule, drink every day, but we 'save up' and go mental at the weekend or special occasions. This accounts for the 'paradox' of how some countries can approach our level of consumption without our reputation for excess - they drink as a civilised, adult pleasure, to be enjoyed in moderation; we drink like it was going out of fashion.
Don't believe me? Want more citations? Here are some public Dublin webcams . Log on to any of them, any of them at, say, 03:00 on a Friday night/Saturday morning and watch the mayhem and madness.
2. Common stereotype: The Fighting (" foightin' ") Irish.
The reality: Ireland is a remarkably violent country.
For example, the U.S. has a violent crime rate of 429.4 per capita .
Ireland has an assault rate of 347.9 for the same period .
"But that's great! Almost ten points less violent than the U.S.! Well done, the Irish!"
Yeah. Until you consider the following two facts:
i. No guns (or very, very few) in private ownership in Ireland.
ii. The Irish figure above is for assault only, whereas the U.S. figure is for all violent crime (e.g. including homicides, excluded from the Irish figure) - the two countries simply do not record crimes in the same way, due to different legislation in each jurisdiction.
Anecdotal supporting evidence:
The [old fashioned] U.S. slang for a brawl is called a "Donnybrook" , after the location in Dublin  where faction fights took place in the 19th century.
Donnybrook was once the location of Donnybrook Fair, a fair held from the time of King John onwards, which became notorious for drunkenness and violent disorder.
Spotting a theme here perhaps?
Anecdotal supporting evidence II: This Time It's Serious.
The common theme is of course alcoholic excess+violence. The inter-relationship here is well established, especially in Ireland, somewhat the "patient zero" for researchers in this field .
- 76% of all rape defendants had been drinking at the time of the alleged offence.
- Alcohol has been identified as a contributory factor in 97% of public order offenses as recorded under the Garda PULSE system.
- One in eleven, or approximately 318,000 of the full adult population, said that they or a family member were assaulted by someone under the influence of alcohol in the past year.
- Almost half of the perpetrators of homicide were intoxicated when the crime was committed.
So the question remains, are we violent when we aren't pissed? This is more difficult to answer, and not only because I don't remember (because I was pissed). Well .. .we've been fighting for our nationhood for 800+ years, although that was hardly our fault.
It is remarkable to note however that when we ran out of foreign invaders to fight we had a civil war . If we'd run out of Irish people to fight, I firmly believe we would have started fights with rocks, the ocean, the sun, and anything else that was looking at us funny.
3. Common stereotype: The Irish all have red hair.
The reality: Ok, obviously we don't all have red hair, I answered on another Irish phenotype here:Domhnall O'Huigin's answer to Who are the Black Irish?
But an estimated 10% of the population  have some variety of the hair that is called 'red,' way above the average. And the highest is Scotland at 13%, and while Scotland is absolutely it's own nation with a proud and unique heritage, the fact is there isn't a lick of difference between the Scots and the Irish; not in culture, ethnicity, etc.
So in short, yes: noticeably more red-heads than lesser countries. So come over, bring your anti-redhead prejudices too, so we can practice stereotype #2 on them, which is to say; you.
4. Common stereotype: Irish people are all fabulously articulate, skilled wordsmiths and poets; just generally wonderful with language basically.
"Oh Oscar, how witty and wonderful with words you are!".
The reality: out of all of the stereotypes in this answer, this is probably the most ambiguous as regards its truth.
On the one hand: Oscar Wilde, Brendan Behan, Samuel Beckett, Edmund Burke, George Bernard Shaw, W.B. Yeats, Patrick Kavanagh, Seamus Heaney, Seamus Deane, James Joyce, R.B. Sheridan, Flann O'Brien, Sean O'Casey, Oliver Goldsmith, Jonathan Swift etc. etc. etc. etc. , .
It is also a fact that as the underdog, and/or new immigrant to a new land, and/or being denied by law our language and education , we took a bardic tradition of fluency and story telling and gave it even more importance in our society, one that survives to this day. Fluency, verbal dexterity, and wit are premium qualities in Irish society in the 21st Century as much perhaps as they were in the 1st.
On the other:
1 in 4 Irish adults has difficulty with reading, writing, and maths. .
That is a pretty damning (and damn scary), not to say shaming, national statistic. Fine, so they are including numeracy there but still.
So don't expect the red-haired, drunken Irish yob who attacks you to always be able to delight you with his lyrical language gymnastics while doing so, as you have a 25% chance of getting the other kind.
Well all the Irish people you have heard of are pretty good with words, true but it isn't part of our DNA or anything - you can be Irish and be rubbish with language - you don't have your citizenship revoked.
In summary, I started answering this question fully intending to debunk all these ridiculous and offensive or lazy stereotypes, and I found out while researching the answer that they were mostly true, so I changed my mind and my answer and decided to challenge the premise instead (admittedly this assumes my common stereotypes are the ones the questioner had in mind - always a dangerous assumption but one I am willing to volunteer you for).
Don't let them put you off us or our country though, please. We really aren't drunk all the time (I'm only half-cut right now, for example), and we usually fight with each other instead of strangers - we pride ourselves on our hospitality (another cliché that is true: the Irish really are welcoming). Just leave your prejudice against red-heads at home when doing so and indulge us in our ways.
Sure we're harmless really:
The whole race is war-mad, says Strabo, high-spirited and quick to fight, but otherwise straightforward and not at all of evil character.
 my emphasis.
"For the Great Gaels of Ireland
Are the men that God made mad,
For all their wars are merry
And all their songs are sad."
[Feel free to disagree - we love an argument!]
 Source: Wolfram Alpha (search for "most violent country"): http://www.wolframalpha.com/
 Table 3.4b: http://www.cso.ie/en/media/csoie...
More questions on Irish People: