You say you bike is a '98 FS600. I don't think the FS supermotard model came out until a couple of years later? If this is correct your bike must either be an FE600 (no e-start) or an FE600E (with e-start) that has been fitted later with motard wheels.
The '98 owners's manual indicates that the FE did not have a battery but you mention you have one. This would indicate your bike is either an FE'E or is an FE that has had some significant electrical modifications done to it.
In the case of the original FE, both the blue stator wire and the body of the regulator is connected to the frame of the bike - I think bolted directly onto it. The regulator is the 1½" sq. x ½" thick block with the yellow and clipped brown wire and it prevents the voltage of the electrical system going too high. Either one yellow stator wire, or both of them paired up together, connects to the yellow regulator wire, then from there goes off to feed the rest of the electrical system on the bike. With this arrangement all the electrics on the bike run on AC as there is no rectifier.
The FE'E (same as mine) also has all its 12 volt stator power connected across the regulator, but now the body of the regulator is NOT grounded to the frame - it is mounded on the plastic rear mudguard to insulate it from the frame. This is because the AC is then fed into the AC side of a rectifier (a 1" sq. x ¾" thick block with four spade lug connections that IS bolted to the frame). The rectifier converts all of the stator output from AC to DC and its negative output terminal is connected (grounded) to the frame. When the DC negative output is grounded like this, neither of the AC input wires can be grounded, hence the mudguard regulator mounting. The DC positive output terminal goes off to charge the battery and run the rest of the electrics, which now operate off DC.
For some stupid reason Husaberg clips the brown wire off the regulator, but it is connected inside the regulator to the body anyway, hence you can run without it by connecting to the regulator body. To answer your question, connect the brown wire on your new regulator to the body of the regulator along with the blue stator wire.
I understand it was later on (Kokusan stator?) that Husaberg started using a more complicated combination of AC for the lights & DC for the battery & starter.
If your bike is blowing bulbs you are probably correct in suspecting the regulator. Mine failed and allowed the voltage to go so high it blew all the bulbs and blew the side out of the battery. It is possible to get into the regulator & fix it, but it is not particularly easy.
Like Sparks, the blue wire going to the positive of the battery has me screwed? It might indicate your electrics are not standard. Get back for sure if your bike has an e-start.