Dia daoibh!
Long time it's been! Things get crazy though during school...
Anyhow, I wanted to clarify something.
In and around Dublin, I noticed that Dublin itself is baile atha Cliath, but things like "Dublin bus", become "Bus atha cliath". Why is the baile dropped

Go raibh maith agaibh!

A

Dublin can be Called Baile Átha Cliath or simply Áth Cliath. Baile simply means Town so Bus Baile Átha Cliath is just as valid as Bus Átha Cliath the difference being Dublin Town Bus and Dublin bus. Bus Átha Cliath more accurately describes the county of Dublin it serves in my opinion as opposed to just Dublin city.

Most of the road signs refer to Dublin as simple Áth Cliath which just means Hurdled Ford as opposed to Baile Átha Cliath - Town of the Hurdled Ford

In Modern spoken Irish it's actually:

Blá-Cliath. Some people pronounce it with the á nasal, which is quite rare now.

Also there's an interesting sentence:

D'fhágas Baile Átha Cliath ar an mbóthar aici. = I left her in the lurch.

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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

I think it is actually more like Bleá Cliath with a slender b and l (at least in Connemara):

Baile
BUH-lih
/bal´ə/

Átha
AW-huh or AW
/ɑ:hə/ or /ɑ:/

Cliath
KLEE-uh
/k´l´i:ə/

Baile Átha Cliath
bi-L(y)AW KLEE-uh
/,b´l´a: 'k´l´i:ə/

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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 think it would be bus bhaile...

Right you are! Bleá Cliath in Munster as well.

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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

Grand along the lines of what I was guessing and extra! I also saw it spelled Ath Cliath today.

Maybe im wrong but I always assumed (and always used them as such) that Áth Cliath was used to refer more to the county as a whole while BÁC was more so for the city, the dublin GAA team uses Áth Cliath which makes sense to me 'cause they dont just represent the city, and Dublin bus goes to places which would be considered outside the city limits, even goes over the border into Wicklow

That was my feeling yeah, refering to the county simply as áth cliath, although I have seen it one or twice as Contae Bhaile Átha Cliath, I think most people take Baile Átha Cliath just to refer to the city.

Ó 'brón orm a Somhairle Óg, im only after realising i basically just repeated your answer, ill have to read all the posts mare carefully next time, gabh mo leithscéal a mhac