I've found the key to great images with a regular compact/point-and-shoot can be as simple as getting the lighting right! Situations where the flash can be firmly off usually work best for me.
examples (Canon IXUS 900Ti):
https://images-1.redbubble.net/img/art/border:blackwithdetail/product:laminated-print/size:medium/view:preview/520698-3-alone-in-paradise.jpg
https://images-0.redbubble.net/img/art/border:blackwithdetail/product:laminated-print/size:medium/view:preview/510273-3-i-see-you-watching-me.jpg
and this was taken on my phone (sony ericsson k800i - 3mp)
https://images-1.redbubble.net/img/art/size:ularge/view:main/977666-2-still-hooked.jpg
not claiming to be an expert - just letting you know what seems to work for me!