I'm not too sure on the popularity of the spelling of Wrenne in Ireland but I know that when my gg-grandfather left Ireland and settled in Canada his name was spelled in a variety of ways - Wren/Wrenn/Wrenne. It's hard for us to know what the actual original spelling was, we can only go by the documents we've found here and as I say, every variation imaginable is seen in the spelling of the name. Many of those who emigrated to Canada were unable to read or write so they were at the mercy of the person compiling the documents and probably didn't correct anyone on the spelling. There might be many original Wrenne members now using a different spelling in North
America.
Regards,
Frankie McGraw-Nicol