While my instinctive reaction to Heroku's shift was "ugh, that sucks, how dare they," I have to say I get it. It is hard to build a successful business with that model given their offering. However to the comment about "git push heroku master," someone that is an "early" programmer like myself finds that level of simplification helpful when I'm focusing on things like learning all the other things that could go wrong…
> I'm wondering if there are any recommendations for "Heroku-like" providers with a free tier that stays up 24/7 for super simple Rails and Sinatra apps, perhaps with a DB. OpenShift( https://www.openshift.com/ ) may be the service you're looking for. I haven't tried it, but it looks nice.
I've only used it to create a small Python/Flask app http://wwos-bitmason.rhcloud.com/ but there are lots of examples and information on the OpenShift blog.
[Disclaimer: I work for Red Hat.]