I’m having the exact same issue in IE10, Firefox 18 and Chrome 24. If I click a button to change the month, year, or even layout, nothing happens. If I then refresh the page, it then changes to what I asked.
I’m using the latest version of WordPress, TwentyTen (child) and All-in-One Event Calendar. Disabling all other plugins and using the TwentyEleven theme doesn’t help. Definitely seems like a bug in All-in-One Event Calendar!
Website is here: https://www.jimal-khalili.com/calendar-2