I think the standard term is Mamaí, vocative a Mhamaí.

In Connemara Irish it varies from family to family and includes:

Mama, vocative a Mhama
Maime, vocative a Mhaime
Maimín, vocative a Mhaimín

And "Mum/Mam/Mom" is Mam, vocative a Mham.

What forms are used in other dialects?

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

I've heard Mamaí, Mam, Máthair, with the last two being the most common after the kids
are a bit more grown up. Also Mam still has the full case system (nom. Mam, gen. Maime, dat. Maim) for
some people as I have been corrected from "lem' Mham" to "lem' Mhaim". It's pronounced Mawm, like the
way an American would say it in English.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

Brendan, how is Maime, vocative a Mhaime pronounced-- or does that vary as well? I've heard it pronounced "mee-mee" and "a wee-mee," and "mummy" and "a wummy," both by Connemara natives from the same area.

Any thoughts?

Thanks!

I'd say "mummy" /mam´ə/ and "a wummy" /´ə wam´ə/, though the a tends to be lengthened /ma:m´ə/ (first syllable like "marm" with a New England American accent, i.e., without an r"?) and /´ə wa:m´ə/.

@AnLonDubh - Interesting. Is the vocative of Mam still a Mham there?

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

How could I forget the most important case of all for this word!

Indeed, A Mham. In Cork (not Kerry) they sometimes treat feminine words as if they were masculine
in the vocative, so it's possible that somebody might say A Mhaim, but I've never heard it.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

In Ireland everyone from the country calls their mother "Mam" or Mammy (when they want something ), this "mummy" is directly from English telly, if you watch Eastenders or Cornation Street. I have only ever heard city people or well to do people call their mother "mum" or "mummy". Lately in the Irish media you hear "mummy" and I'm sorry but its one of my pet hates . Funny enough these "mummy" Irish people will often indirectly address their mams or mams in general as mum, but when the are speaking directly to them they call them "mam" . People from Dublin often call their mams and dads "ma" and "da".

Back to the Irish:

a "mhathairín" as well as those Lon Dubh gave is used quite a lot in Munster especially when children become teenagers. I have heard both "a Mhaim" and "a Mham" for the vocative.

I've heard the reason Americans call their Mams "Mom" is because of the close resemblance in pronounciation to the way Irish speaking people pronounce "mam". Don't know if that's true or not.

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I am sure it must seem that way from your corner of the world, but "Mummy" is actually the standard spelling everywhere except the US, and "Mammy" is only used in Ireland. (So people searching for the answer to that question are more likely to be looking for "Mummy" or "Mommy".)

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

I was aware of that alright ,there are a few Aussie shows airing here lately...Master Chef Australia and World's Strictest Parents being some; and they always call their mothers' mum too, so I kinda guessed that.

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In the children's book 'Ulchabháin Óga' by Martin Waddell that I've just bought today - it says not only Mamaí but Mumsaí as well, and A Mhumsaí in the vocative.

Cringe, , That's as bad for me as someone saying "Ó Mo Dhia" . Mumzy is what really posh Englih people call their mother as a pet name!

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