Formality of 'but' at the end of the sentence?

Sentence-final but as it is known is a well documented feature of English as spoken in Ireland, much of Scotland and also in Australia and New Zealand. Although it is perfectly grammatical, it is not considered formal and won't be seen in formal writing or documents. It has been the subject of many academic papers in linguistics and a few books. If you'd like to read up on it, here's a source to get you going:

• Mulder & Thompson, 2008, 'The grammaticization of but as a final particle in English conversation' in Ritva Laury Studies in clause combining. John Benjamins. pp. 179-204.

You will hear this in the parts of the Midlands and north of England, possibly Wales, but you will not hear it in the South. There, you're more likely to hear 'innit' or 'bruv' than 'but', 'innit' in particular being an even more pointless addition to a sentence - generally, in parts of the country where the use of 'but' at the end of a sentence does not occur, it can cause confusion or be thought of as quaint but peculiar. Geordies in particular often use but in this way, as in "she's a canny lass, but", where it doesn't mean anything other than a full stop. Certainly, neither would appear in formal print.