Unless otherwise expressly provided, no motor vehicle shall be operated or parked and no trailer shall be towed or parked on the highways of this state unless the motor vehicle or trailer is registered in accordance with the Motor Vehicle Registration Act. There shall be a rebuttable presumption that any motor vehicle or trailer stored and kept more than thirty days in the state is being operated, parked, or towed on the highways of this state, and such motor vehicle or trailer shall be registered in accordance with the act, from the date of title of the motor vehicle or trailer or, if no transfer in ownership of the motor vehicle or trailer has occurred, from the expiration of the last registration period for which the motor vehicle or trailer was registered. No motor vehicle or trailer shall be eligible for initial registration in this state, except a motor vehicle or trailer registered or eligible to be registered as part of a fleet of apportionable vehicles under section 60-3,198, unless the Motor Vehicle Certificate of Title Act has been complied with insofar as the motor vehicle or trailer is concerned.