Strings to a bow or jack of all trades and master of none?

Having been made redundant a few months ago from my first professional graphic design role I'm once again on the job hunt. I know this isn't an unusual tale at the moment so I wondered what all your opinions were about design skills and how they are perceived by a potential employer.

I'll give a bit of background about me: I've a degree in Illustration with Animation, worked in video and post production, looking for a job in graphic design and occasionally dabble in web design. I enjoy learning new skills and always thought that if I have more to offer an employer the better ie. the more strings to my bow the more employable I may seem.

But it got me thinking - is this approach seen as unfocussed? Am I being seen as a jack of all trades master of none? What do you think? Is it good to have as many skills as possible or is it better to be super fantastic in every way at just one thing....?:icon_confused: eek!

Job adverts will usually list certain skills as essential and others as desirable: stress the essential, mention the desirable and forget the irrelevant.
Decide where your passion is, work on that and use your knowledge of the rest as nothing more than an 'understanding'. If you've done everything in fits and starts, it's hard for an employer to believe you're going to be dedicated to them but if you can show you're passionate about a particular discipline AND understand how the rest of it works, you've instantly got the edge on other applicants.
Thanks for those replies! It's good to get a different perspective, especially about the dedication part. I think my main problem is the actual getting experience in one area, hence the fits and starts approach! Any suggestions?! I am using a lot of online agencies, however, can't seem to find any that are particularly design based....
If you're going to focus on one area, I would say polish up on web design (not necessarily the code as I know many great designers that just pass it on to a developer as roles get more specialised) - this is the major money for me currently as a freelancer, and I know a lot of agencies are streamlining to digital.