Unwanted pixelated art when it should look smooth... :(

Hi, I am new to Unity and can't figure out what I am doing wrong. The graphical elements look awful. The screen shot is part of a screen. All things in the blue bar are UI images and the things on the green background are sprites. Any hints to where I should start my trouble shooting?

 

I have done a test to see what affects a png image in Unity. All shoes in the attached image are imported to assets as png, then draged onto the scene. This makes them sprites with a sprite renderer. The image in the middle has no changes but the big one is scaled to 2 and the small one is scaled to 0.5. What puzzles me is that the small image is so pixelated. Normally it is a good thing to use a larger size and then downscale it rather than the oposite. But here the enlarged shoe looks better than the diminished shoe...? Why? And how can I control the outcome of my graphical elements?

 

Downscale to 0.6 - 1.0 can be good
downscale to 0.5 (and less) give you Aliasing (source: https://en.wikipedia.org/wiki/Bilinear_filtering Limitation part)
due to this effect, bilinear filtering always have limits of applying (near 0.5 as I know), and when image must be downscaled more than 0.5 - the mip-mapping comes on, i.e. create 0.5 image prepared (png) and use it where you need to downscale original (1x) near 0.5-0.25 and use this new image instead (you can have 2 copy of it - for Big and for Small instance in UI)
or use mip-mapping in your sprite in settings (imho it will not work, as I know)