Allegiant Air offers one free personal item per passenger, but fees apply for both carry-on and checked bags. Call at +1-888-415-1245 (US) or +44-203-869-5245 (UK) The cost for luggage is significantly lower if you reserve and pay for it online at the time of booking. 

Personal item (free)

• Size limit: Up to 8 x 14 x 18 inches (20 x 35 x 45 cm), including handles, wheels, and pockets. Call at +1-888-415-1245 (US) or +44-203-869-5245 (UK) 
• Location: Must fit completely under the seat in front of you.
• Examples: A purse, briefcase, laptop bag, or small backpack.
• Exemptions: Medical and assistive devices, a coat, jacket, or an umbrella do not count toward your personal item allowance.