Today is St. Patrick's day, which means there's a lot of this online:
Lá Féile Pádraig shona duit
Lá Fhéile Phádraig sona duit
or basically any variation of:
Lá F(h)éile P(h)ádraig s(h)ona duit
which all roughly mean 'Happy St. Patrick's Day' in Irish, but some of which are grammatically incorrect. (Note: I'm not going to get into pronunciation here, but for the curious, here's a video.)
When I was learning Irish in school this was generally the trouble. The meaning is clear most of the time, but where or when to put the séimhiú (lenition - those 'h's above), the urú (eclipsis), or the additional 'i's (slendering) was never particularly clear. I got through Irish on the vague feeling about what was correct, mostly based off the sound of the thing.
On some recent trip home to Dublin I bought a book on Irish grammar in some optimistic hope of fixing the situation. Inside this book must be contained a set of concise and logical rules, the mastery of which will enable me to unambiguously resolve problems like the above. (hahahaha)
Let's break the phrase down. Translated literally (and with caveats)
lá fhéile Pádraig -> the feast day of Patrick
X shona duit -> happy X to you
The second one is a bit easier so we can get that out of the way.
'Sona' means 'happy', and as an adjective it is modified to agree with its noun. As a member of the third declension, 'sona' undergoes no change to its ending, but it does pick up a séimhiú (h) when it's paired with a feminine noun.
'Duit' is a prepositional pronoun, which in this case means
duit = do + tú = to + you = to you
the 'to' of 'do' is in the proposition sense (don't try to use it to form infinitives of verbs, or anything like that).
So we really have
XY sona duit -> happy XY (masc) to you
XX shona duit -> happy XX (fem) to you
Now for the first part. This is where things go wrong for me.
'Lá' means 'day'. It is masculine.
'Féile' means 'feast'. It is feminine.
Tearma.ie seems to think that one makes 'feast day' with 'lá féile', which seems to agree with my grammar book. In the compound noun, 'féile' should go into the tuiseal ginideach (genitive case). As a fourth declension noun, this means it undergoes no change. The resultant word ('feast day') is masculine (the gender is taken from the first noun), so we'll follow it up with 'sona'.
However, tearma.ie also says St. Patrick's day is 'Lá Fhéile Pádraig', so something is going on here. Is the presence of the 'Pádraig' complicating the issue?
The answer that I've found for this is: yes. My grammar book is silent on the topic of this slightly-advanced compound noun case. According to Nua Leargais, when two genitives come together (as we would naïvely try for 'féile' and 'Pádraig' after 'lá'), we encounter the functional genitive. In this case:
What the hell is the genitive of Pádraig?
The answer seems to be 'it depends'. Situations like this are why I end up feeling like 'Phádraig' and 'Pádraig' are equally valid and get confused. It seems like:
In general, the genitive of 'Pádraig' is 'Phádraig' (so 'Patrick's house' is 'teach Phádraig'). This is supported by tearma.ie. However, in special cases it doesn't change, and those cases include 'after féile' and 'after naomh'.
So to summarise, if you want to say 'Happy St. Patrick's Day' in Irish, and be (probably) grammatically correct about it, it's:
And now you know.
 This turned out to be sufficient to get an A1 in the exam (somehow), but I'll never feel capable of teaching anyone the language if I can't explain why things are how they are.
 We learned the prepositional pronouns using songs and hand gestures in school, so they're indelibly seared in my memory. Here's the full list of the 'do' ones:
dom = to me
duit = to you
dó = to him/it
di = to her/it
dúinn = to us
daoibh = to you (plural)
dóibh = to them
And here's a page with some more. You might notice that 'do' means 'to/for' there, which is true, but that is beyond the scope of this post.
 'Naomh' means 'saint', so if you, like me, went to a school named after St. Patrick and grew up hearing 'Naomh Pádraig' all the time, you might be forgiven in thinking that's its genitive form.
Here are some pages I found useful/interesting while trying to resolve this problem: