We are based on OSM, and while not perfect, we are pretty satisfied with it. If you want to give it a try to see how it looks, you can see it here: https://play.google.com/store/apps/details?id=org.bikecitygu... (you can test it for free and if anyone is interested in coupons let me know)
One of the main advantages for us is that we made an in-house routing algorithm so routing is done on your phone, that means that you don't need an internet connection while routing, but for that we need to download the maps, of course this is not possible with proprietary maps like Google Maps.
At the moment we're only in Europe and the map quality has been pretty good. We're are having some problems with some cities, like Madrid with the direction of the streets, but you know what's awesome, that when we get a mail from a customer telling us that this and that is wrong, we can reply them in a couple of hours telling them that the issue was fixed and it will be deployed in update.
I do use Navit [1] offline navigation which I belive uses offline maps based on OpenStreetMap. GUI is a bit quirky, but it works. Offline maps are downloaded from the application. I use simply snapshots apks from Navit's download page [2].
I've encoutered some bugs in the maps for new build roads. New means few weeks, but it otherwise works very good. It can be used as car navigation because energy consumption is rather fast so connecting to a car charger is a must.
[1] http://www.navit-project.org/ [2] http://download.navit-project.org/navit/android/svn/
Another app that has been praised by a lot of people is CityMaps2go [0], both on Android and iOs. While the application is not open-source (which is why I don't use it), the underlying data comes from OSM, so by using this application you're fighting for OSM.
[0] https://play.google.com/store/apps/details?id=com.ulmon.andr...
Indeed, the navigation works great IF you can remember how to work the UI. Never the less, I'm still in awe as how well accurate OSM is (in the Netherlands + Belgium).
The UI is not perfect, but it has lots of useful features, like bike navigation and offline usage.
When I finally get the results of my query for "Hammersmith" it tells me that 309km away (some mistake, more like 3.9km) there is "Subway Station District Line;Hammersmith & City Line", and then the same result repeated around 150 times some with slightly different naming, and at varying distances from 309-328km away before it gets to other results in Hammersmith.
And you are trying to use this on a phone in the rain, you just give up and go back to google maps even though the data is worse.
EDIT: And if you select one of these it appears to be in Barking, as it found every station on the Hammersmith and City Line, although that is not 300km long.
I'm not sure how it could remain as featureful as it is and improve the UI. So I'm quite happy with it.
The number one huge win for Osmand is offline searching for points of interest. Being able to to pull up a list of all playgrounds close to where I am right now is invaluable when traveling with a hyperactive 3 year old.
I'm guessing there's a bunch of other niche apps, that do the same kind of thing (e.g. I'm vaguely aware of Wheelmap for wheelchair users).