TL;DR (aka executive summary)
- Yes, you should be worried.
- Yes, this is severe (giving total strangers potential complete control over your files and resources).
You should definitely upgrade your desktop AS WELL AS any servers.
Your DHCP client uses
dhclient-script which uses shell variables passed from the server. If there's a rogue/compromised router, it may pass modified
domain-name variables with the exploit.
Credits: Stéphane Chazelas, Mark, Michal Zalewski
In addition, many desktops use OpenSSH, which is definitely vulnerable as per http://seclists.org/oss-sec/2014/q3/650 (although for logged-in users - aka rogue insiders on your network). According to the original reporter of the bug, Stéphane Chazelas, the OpenSSH vulnerability is about bypassing ssh
Please note, however, that a full fix for the issue is not available yet (http://seclists.org/oss-sec/2014/q3/679).
See https://access.redhat.com/articles/1200223 for possible workarounds.
Debian hasn't published the upgrade yet, and I haven't had the time to find a relevant discussion on their site.
(NB: Second a suggestion by terdon; it would be very nice if Stéphane Chazelas could write down a canonical Q&A on Shellshock.)