If you wish to fly within the Class B or C airspace, then you are required to have the transponder whether or not you have electrical, single seater or no.
Lack of electrical system excuses you from needing to have a transponder within the 30-mile Class B "Veil", as long as you stay out of the actual Class B airspace (inverted wedding cake). Again, doesn't matter how many seats the plane has.
A single seater *is* excused from the ELT requirement.
Ron Wanttaja
Well, you can fly within B or C if you have an aircraft that is not equipped if you make the request at least one hour prior to the flight. See 91.215(d)(3).
Anxiety is nature's way of telling you that you've already goofed up.