Catalonia: Catalonia has been autonomous, a part of Spain, autonomous again, and invaded, suppressed, and held since about 1469, but, most recently, has a rough majority to form it's own nation. The Spanish parliament just recently voted to stop a referendum on Catalonian independence, however, and this seems unlikely to change.
Scotland: Scotland has been in the news of late with the Scottish National Party holding a majority in the Scottish parliament, and planning a referendum on Scottish independence in 2014. While they do not desire to be a part of the United Kingdom, they do not necessarily see themselves as a republican organization, and do not seem adverse to keeping Queen Elizabeth II as their monarch.
Quebec: This is perhaps the one I am most familiar with, simply due to it's relative proximity to my own home. Quebec has had the Parti Québécois since the 1960's, and has had varying levels of support, calling referendums in 1980 (rejected by 60%) and 1995 (50.6% "No" to 49.4% "Yes"). In 2009, when asked "Do you believe that Quebec should become a country separate from Canada?" 34% replied yes, 54% said no, and 13% were unsure, though in that same poll, only 20% believed that Quebec would ever become a separate country.
Will independence ever happen for these groups? I tend to doubt it. With these smaller nations in particular, I'm not certain it would even work out all that well financially. And yet, it is plain enough that there are sizable portions of people who want to be autonomous, and perhaps with good reason too.