I like your definition of "slight"
Seriously though, the answer probably depends on whether you are rooted and whether you have usb debugging switched on. If the answers to both of those are yes then you should be able to "adb pull" the message database off the phone.
The problem otherwise is that the messages will be in a database in the /data/data area (/data/data/com.android.providers.telephony/databases is the standard folder), and you'll need root to view or copy it. And I doubt that area will mount on the computer anyway, in which case you'd need to use adb to pull the file, which is where usb debugging comes in: if that's off it's going to be very hard to turn on without the screen!
If you could run a backup app you could copy the messages from the backup, but the problem is running it without a screen.
That's one reason I like SMS Backup+ - you set it to automatically back up and you have the backup in your GMail account, off the device, if anything like this happens (less essential for me now that Titanium can back up SMS in a portable format, but still a nice insurance). But no use unless you were already running that as it needs setting up.