Dubbed laughter was used for the first time on the "The Hank McCune Show".
OR
Dubbed laughter was used for the first time on the "The Hank McCune Show."
Where should the fullstop come?
At the end of the complete sentence: Dubbed laughter was used for the first time on the "The Hank McCune Show".
Logical or not (and it's often not), the US style calls for full stops to always go inside the quotation mark. It's one of the few "always" rules I quote, perhaps because it annoys me so much.
If you're writing for an American audience, put it before the quote. If you're writing for anyone else, it's as 5jj says.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
Don't forget that "full stop" is two words, not one.
That's what I was taught for direct speech many years ago: John said, "I enjoy the Hank Mcune Show." As far as I know, some guides still recommend that. Logically, It should be: John said, "I enjoy the Hank Mcune Show."., but that does seem excessive. I now go for: John said, "I enjoy the Hank Mcune Show". This is the best compromise, for me.
For a title in quotation marks, I think it was always acceptable, in England, to put the full stop at the very end, as it is not part of the title - Dubbed laughter was used for the first time on the "The Hank McCune Show".
Learners should note that we are not too fussy about punctuation in the United Kingdom, with a few exceptions (such as with defining and non-defining relative clauses) so long as it makes reading easier. If you are writing for publication, the publishers will punctuate in line with their house style, whether you like it or not.
You can avoid the issue by using italics:
Dubbed laughter was used for the first time on The Hank McCune Show.