These are pretty much the only scenarios where antibiotics should even be considered, however, you still can resolve these issues without. Aggressive sinus rinsing (3-4 times daily) with either a Neti Pot or NeilMed squirt bottle is imperative. You can also add some antimicrobial compounds (peroxide, herbs, tea tree oil, etc.) to facilitate clearance. Hydration of 3-4 liters of water daily will help. Avoidance of gluten, dairy, and sugar for 2-3 weeks can also help. In the early days of the infection, regular zinc intake can help resolve it prior to escalation.
There are a few unique situations like imaging diagnosed sphenoid sinusitis that should always be treated, but that is rare. As a primary care physician, I've had a number upper respiratory and sinus infections over the past 20 years and never taken antibiotics once. I typically immediately start Zinc 30 mg twice daily, sinus irrigation 3-4 times daily, lots of hydration and avoidance of irritant foods, and eat lots of mushrooms (immune support), and that does the trick.