When I looked at them, I chose [Edit: correct name Advanced Ruler] from Ying Wen Technologies--features include sliding cursors, calibration of scale on screen and calibration to edges of phone (or phone with case), built-in database for sizes of screens on lots of common phones so you don't have to calibrate to be up and running, circle size tool, cm or inch scales, and it looks nice, like an engraved metal ruler, for whatever that is worth.
Note that there are also some camera-based apps that triangulate using the angle sensor to measure distances to far-away objects, or heights of them, using your height (which you calibrate) as a basis. Advanced ruler pro ($2.99) adds that feature but there also also free apps that do that--I have "smart measure".
While you are equipping your phone with that stuff, consider a bubble level as well. I like both "bubble" by ben zibble and "bubble level" by antoine vianey installed--each have features the other doesn't have.