How to get my post title to work with an apostrophe (‘s)?

EDIT:

Disable magic_quotes_gpc in your server.
try adding in .HTACCESS file (if you on shared hosting):
php_flag magic_quotes_gpc off
If you’ll get 500 server error after you added it – delete it and put this:
ini_set ('magic_quotes_gpc', 0); in theme’s functions.php file. And with function bellow check is it on.

You can create php file with this content to test is it on (if on you’ll see this nice string):

if (get_magic_quotes_gpc()) {
    echo 'F@ck.. its on :D';
}

2nd try:
If the magic quotes was ok, than MAYBE you can try and search in your plugins, and theme’s php files (with textcrawler) do some of them uses stripslashes function on title?