How to find web hosting that meets my requirements?
I'm looking for free hosting with HTTPS and PHP support. Of course a subdomain is enough. Something like Google's appspot.com, but with PHP support.
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Hosting does not tend to cost to much if you shop around. and if it is for a small traffic sight or just to get people to view your page you could try self hosting. I do that when I need to test php scripts or develop sites before I have proper hosting setup. I use XAMPP from apache friends. It has simple setup for an apache server and supports PHP5 and MySQL right out the gate.
PHPFog is a 'cloud platform for PHP' with a free (for six months) plan called "Shared Cloud" that comes with shared SSL for HTTPS support. It's limited to 100MB of storage and 15GB of monthly bandwidth, but offers upgrade paths to their paid services if you outgrow that.
Finding free reliable web hosts in general is tricky. If you care about your Facebook application, you may also wish to look at paid (but cheap) alternatives. Many of the large shared web hosting companies offer PHP with shared SSL certificates. (e.g. Hostgator's hatchling plan from around $4/month).