I am working on an application in the Mobile/VOIP domain. This is really a gray area for me. Here are some details about the application:
I am one of the creators of Padrino but I have also worked extensively with Rails and Sinatra. Probably not what you want to hear but no matter what you pick, you will be able to get this project finished fairly easily. I can't say picking one will impact you much over picking any other in the grand scheme.
I am obviously a proponent of the modular and lightweight nature of Rack and Sinatra. Between Rack, Rack Middlewares, Sinatra and extensions, you can get anything done just as easily as in Rails if you are willing to understand the tools.
I would argue that Sinatra and Padrino have a lower learning curve to Ruby newcomers. This is because they promote "take what you need" and "gradual complexity" far better than the more "take it all at once" Rails approach but on the other hand Rails has much more documentation, blogs, support, etc. So the trade offs are clear. Sinatra and Padrino are also much "faster" and "lighter" in terms of memory footprint, requests per second, cpu usage, etc but Rails is fast enough for most situations and the application server is rarely the bottleneck anyways.
All that said, I will try to give you a more direct opinion. If you are doing nothing but a service API (which it sounds like here), I would recommend using Sinatra, Padrino or even another project of ours Renee over Rails. Rails is overkill for a lightweight service API by most measures.
Narrowing it down further, Padrino is Sinatra so you don't have to choose between them. You can start with Sinatra and include standalone modules from Padrino, or use a full-stack Padrino application which is still Sinatra under the hood with a very minimal performance penalty for access to a lot of powerful features (i18n, logger, admin panel, caching, generators, form helpers, mailer, etc). Keep in mind these are all "take them only if you need them" modular extensions.
I would recommend checking out our Padrino Getting Started guide for a place to start exploring Sinatra and Padrino. Our Padrino guides and documentation strive to be thorough. That said, the "safe" bet is Rails since it has a lot more usage, it is more mature, has more contributors and a lot more documentation / googleability. Good luck, hope this was helpful.