Thank you for your suggestions!!
Dublin: The Cobblestones on at the top of Smithfield and O'Donoghue's on Merrion Row. You can sometimes also find a bit of a session on in O'Neill's on Capel St. Now, if you're really lucky and happen to be in town on the second Sunday of each month (for example July 10) there is a singing session in Chaplin's Bar on Hawkins St. These singing sessions are a real treat!
Enjoy.
Not sure what you mean by 'music sessions'. If you mean traditional music in pubs I can definitely recommend the Tig Coili pub in Galway - http://www.galwaycitypubguide.com/tig-coili.html. We spent a wonderful evening there many years ago but I'm sure it's still going strong.