First of all you have to 'Activate' the home.php to be able to use it.
To do this goto Dashboard -> Pages & Themes -> click on the Inspect button beside your theme. You should see the home.php with a check mark beside [] Create page type. Just click on Activate files and the page type for the home.php should be created.
Once done, change your Home page design/page layout to Home page type.
No need for a single page what so ever.

Thanks for your quick reply. I followed your instructions. Unfortunately the result is the same. My home page doesn't show up.

Then I tried to install a new theme (to save mine...) in fact a copy of 'Green Salad' (because this theme has a home.php file and belongs to the original package supplied by C5). I copied it into c533\themes\greensalad2. Then I modified my home.php (remove the left col) and add 'HOME' after the string 'Built with concrete5' (to be able to see wich php page the system uses). Did the same with the default.php (add 'DEFAULT'). Go to Themes -> Page Types and activate the Home page type. You can see the result athttp://www.veritassearch.com/c533/.... The home page doesn't use the home.php file, but the default.php. Why??

C5 Version: 5.3.3.1.

Did you put the home page in edit mode and click on the 'Design' tab at the top, select the 'Home' page type under 'Choose a Page Type', save, and publish your edits?

Thank you friend! It perfectly works!

After activating a new page type I don't see it available. Is this information still the same a year later?

On creating a custom home page:

Just a message out there to anyone who is not too gifted like me, I was trying to create a new custom home page rather than change the existing home page's theme to be that of home.php.

may save someone a few minutes or more of head scratching!

nicko.me

I just wanted to say thanks as well. I was working on my own theme and I just wanted to summarize the process in a few clear steps for Concrete5 5.5+:

1. Create home.php directly in the theme folder.
This is just the template file but to be used it needs to be converted to a Page Type.

2. Visit "Dashboard > Pages & Themes > Themes"

3. Click Inspect next to your theme. You should see the template home.php.

4. Ensure "Create page type" is checked for the templates in question and Click OK to create a Home page type. You can follow this same process for other page types too.

5. Visit your homepage, switch to Edit Mode. Under "Design > Choose a Page Type" select Home to convert the page to the new Home Page Type.

Adding additional templates & page types works the same way:
Create .php template -> Convert to Page Type -> Create pages with the type

@erikig, thanks for this update on 5.5. I was starting to get a bit frustrated.