The story is now coming to a point where you are made to wonder who will take charge of the Akatsuki. It is of course going to be one of two, either Nagato with his controlled bodies of Pein or Madara so really its a battle between the best Sharingan user and the best Rinnengan user. So, which is more powerful, Rinnengan or Sharingan?
We only know a little about the Rinnengan so we assume it only has one state and in that state its said to be the most superior of the three Ocular Kekkai Genkai. We do know that apparently the first user of the Rinnengan, the Sage of Six Paths is told to have founded the ninja world and know all ninjutsu. Also, something probably not worth mentioning is that he apparently created the moon with the technique Chibaku Tensei. Along with Chibaku Tensei, it is thought that BanshM Ten'in and Shinra Tensei are among the techniques that come with possession of the Rinnengan, techniques which basically grant the user to control gravity and push / pull objects from their own center.
The Sharingan on the other hand has three main states, normal Sharingan which gives the ability to read movements, copy jutsu and cast basic genjutsu.
Then there is the Mangekyo Sharingan, acquired by killing somebody close to you. Abilities depend on the user but so far we've seen the Mangekyo giving abilities such as powerful genjutsu and space / time distortion. Examples are Itachi's Tsukuyomi which casts the victim under genjutsu and enables the user to control time along with everything in the created world and Kakashi's only know Mangekyo technique which gives him the ability to distort space and transport objects to other dimensions. Black flames called the Amaterasu that burn until it consumes the target is another technique used by Itachi. The downside to the Mangekyo is that the user ever ends up blind if they overuse it which brings us on to the next Sharingan level, the Eternal Mangekyo Sharingan.
The EMS is admitted when an Uchiha with Mangekyo receives another Uchiha (who also has Mangekyo) members eyes, this merges the two patterns of these Mangekyo to make one and gives the user 'Eternal Light' meaning the user will not become blind if he uses the Mangekyo's abilities a lot. We are told that with the EMS comes many new Mangekyo jutsu so we can only imagine what other abilities Madara has since we've already seen him make any jutsu pass through his body.
From what we've seen I'd like to think that the EMS would await over the Rinnengan and without Naruto kills Nagato in the coming manga or the story suddenly ends then I hope we will see a Sharingan vs Rinnengan battle sometime soon!