Recommended oil viscosity for general riding and all temperatures is: SAE 10W-40. We recommend adding 2oz. of Zinc ZDDP additive per quart of oil with every oil change.
We've also had success running diesel truck oil at 15W-40 viscosity. Diesel oil has the bonus of extra detergent additives. Synthetic oils are are not recommended as they will cause your clutch to slip. As of late 2023, Shell Rotella T4, Chevron Delo 400 and Mobil Delvac 1300 are now a semi-synthetic oil and cause clutch slippage in these engines. Be sure that any oil you are using is still 100% mineral oil.
Should I run a newer style synthetic oil on my vintage Honda?
No, synthetic oils will cause the clutch to slip.