The easiest way of dealing with NoSQL databases in azure without and hassle, out of the box scalability and easy management is using Azure Table Storage.
Some people have often used MongoDB for data storage although, the replica sets of it can be a little tricky to deal with. Creators of MongoDB have now formalized this to run as a standalone server in an azure role.
Any other NoSQL database can also run on it as far as azure allows an unattended install or you are able to either xcopy-deploy.
An explained blog on Azure Data Storage will also be helpful to understand other ways to use NoSQL services.
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