I'll answer your second question first since it affects the first question.
Honestly, I don't have an answer regarding how you should handle the resignation. If they can fire you without cause, then there is nothing you can do. If they have to have cause, you can fight it or try to go to court. If you fight it, you could end up being miserable there as they will make your life hell. You could sue, but that could take a long time and you may end up losing. However, since the job market has tanked, who knows how long it will be before you get another job. Based on the brief details you provided, I would resign since you may not like working there. You could also ask for a transfer to another department if that's possible.
Now, if you do resign, you can simply state on applications you resigned to pursue other job opportunities. This is true. However, if any prospective employer asks, they may find out about the issue between you and your former supervisor. You can state that there were difficulties between you and your supervisor, so it was best to move in in the best interest of the organization. This makes you look good, as you put the organization ahead of yourself.