I, personally, like to go by Briggs & Stratton's recommendations for any small lawnmower engine. It seems to be working fine for me. Their #1 recommendation is SYNTHETIC oil, in 5w30 or 10w30, for any temperature range. They do not recommend dino multigrades, saying oil consumption may result. They sell, under their own brand name, a 5w30 synthetic. They also recommend and sell 30w dino, for use in warmer temperatures.
I have a 2000 model Craftsman mower with the 5.5 hp Honda OHC engine. I have an hour meter on it to track usage, and it is used a little over 50 hours a year. After a 5 hour break-in, I used Mobil 1 5w30, changing once a year. It never used a drop in 4 years. This year, I had a quart of Mobil 1 0w30, so I used that. This oil now has 29 hours on it, and no usage as yet. The oil stays very clean, and I was tempted to use last years' oil two years. I wanted to see what the 0w30 would do, so I went ahead and changed.
I am convinced that lighter oils work best with the splash lubrication systems in most of these little air-cooled motors. Better distribution of oil when cold, less drag for a little more power and fuel efficiency. Heck, if I see no usage with the 0w30 this year, I may try 5w20 next year.