Insert row into database using values from a html form submission
With prepared statements, I don't see anything that would immediately make it susceptible to sql injection (this isn't to say there isn't something that is there at a deeper level - I'm not a security auditor nor intimately familiar with php).  The "or something else" part remains quite large.

Look at OWASP (Open Web Application Security Project) which goes through a large set of possible attack vectors through the web.

Such a page is susceptible to various scripting injection where someone inserts a script (or flash, or even something like an image from another site) to be executed on another client machine. From the OWASP top 10 - 2013 pdf the XSS attack can be seen and described on page A3 (page 9).