For men, part of my "hiking pants" have zipped pockets that are big enough and placed well enough that the phone doesn't bother you when running or climbing or whatever. (Don't get me started on women's hiking pants, the missus buys hers from the men's section.)

For shorts: https://www.uniqlo.com/us/en/products/E483295-000/01?colorDi...

They're overall a pretty common product category, though the highest end brands (lululemon) don't have pockets.