Constitute.org is a useful website designed by political scientists to let researchers search for and compare constitutional texts on particular topics. Here for instance is a search on secession clauses. Although one of the site’s creators, Zachary Elkins, says that 22 states contemplate some process for state divorce, only three constitutions expressly authorize some part of the country to secede: Ethiopia, Liechtenstein, and St. Kitts and Nevis. Ethiopia lets each people or nationality secede by a supermajority vote of its legislature, Liechtenstein lets each commune secede (I believe this was an addition of the 2003 constitution), and St. Kitts and Nevis lets Nevis secede by a supermajority referendum vote.
In addition to these, Britain’s Northern Ireland Act of 1998 lets the majority of Northern Irelanders decide to join the Republic of Ireland, and the constitution of Uzbekistan lets Karakalpakstan secede with the consent of the Uzbekistan government.
It would be interesting to see how many states define themselves as “indivisible,” thus tying a government’s hands and preventing it from authorizing secession. A search on the term brings up some irrelevant cases, but 72 constitutions contain the term.
By the way, the Prince of Liechtenstein is a moderate libertarian, and their constitution is fairly consistent with the philosophy. Check out his book on the topic.
PILEUS 
Secession may be a viable alternative for those other countries, but does anyone realize what would happen if the states in the U.S. decided to secede? A likely scenario can be found at http://townhall.com/columnists/kurtschlichter/2014/12/15/nothing-succeeds-like-liberal-secession-blue-america-without-red-america-would-be-a-basket-case-n1931175.