sageRegina is now just regina
sageRegina is now superseded by regina_wheels and available as just regina on pypi.
Thanks to Nathan Dunfield and Marc Culler for automating the wheel building process.
sageRegina
sageRegina is a packaging of the triangulation software regina making its installation under SageMath as easy as running the following command in Linux:
sage pip install sageRegina==6.0.1 log reginaInstall.log
Or typing the following command into a terminal under Mac OS X (without OpenSSL):
sage pip install http://sageRegina.unhyperbolic.org/sageRegina6.0.1.tar.gz log reginaInstall.log
This will retrieve regina from pypi, respectively, unhyperbolic.org and compile it in one to several hours and install it into a subdirectory of the one that sage is installed in. Regina is now ready to be used inside sage. Just start sage and type, for example:
sage: import regina
sage: t = regina.Example3.figureEight()
sage: print(t.homology())
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sageRegina only requires a working sage installation and, on Mac OS X, the xcode command line tools. All libraries that regina depends on are either part of sage or included in sageRegina. sageRegina does not provide regina's GUI. Its installation is modeled after that of SnapPy. The license of regina applies.
Remarks:

Python 3 (and SageMath 9.0 or later) users should use version 6.0.1 of SageRegina. Python 2.x (and SageMath 8.x) users should use version 5.1.5 of SageRegina.

Regina is also part of Nathan Dunfield's Sage Docker image. Docker is native to Linux, but there is a version for Mac OS X available.

Mac users only: You can use the above installation command given for Linux if you run Marc Culler's fix sage script or manually install OpenSSL for Sage (also required to install SnapPy from pypi).

You might need to give the (full) path to sage, either because your system cannot find sage (or because you want to install sageRegina in your own copy of sage when there is also a systemwide installed sage).
For example, on Mac OS X, you might need to type /Applications/SageMath9.2.app/sage.

If you previously installed sageRegina and want to upgrade to a newer version, add upgrade after install to the above command line. You can also use upgrade to "downgrade" to a specific version, e.g., sage pip install upgrade sageRegina==5.1 (see versions).

Writing to the directory where sage is installed might require super user/administrator privileges. In this case, you need to prefix the installation command by sudo.

Alternatively, you can instead add user to the installation command so that regina will be installed as a user site package in your home directory instead.
How do I test that the installation worked?
You can check that sage is picking up the intended version and not an old one by typing inside sage
sage: import regina.sageRegina; regina.sageRegina.version
You can also run the python part of regina's testsuite by typing
sage: import regina.sageRegina; regina.sageRegina.runTests()
How do I install the xcode command line tools (Mac OS X)?
xcode is available for free through the AppStore. Once you have installed xcode, type the following to make the xcode command line tools available:
xcodeselect install
Please report the results if you tried it out on other platforms to Matthias Goerner (enischte at gmail dot com).

Mac OS X

Version 10.14.6 (Mojave)

Sage 9.2: ok (requires sageRegina 6.0.1 or later)

Sage 9.0: ok (requires sageRegina 5.1.9 or later, using sage9.0OSX_10.11.6x86_64.app.dmg)

Sage 8.9: ok (requires sageRegina 5.1.5 or later)

Sage 8.8: ok (requires sageRegina 5.1.3 or later)

Version 10.12.6 (Sierra)

Sage 8.6: ok (requires sageRegina 5.1.3 or later, using sage8.6OSX_10.14.2x86_64.app.dmg)

Sage 8.3: ok (requires sageRegina 5.1.3 or later, using sage8.3OSX_10.11.6x86_64.app.dmg)

Sage 8.2: ok (requires sageRegina 5.1.3 or later)

Sage 8.1: ok (requires SageRegina 5.1.2 or later)

Sage 8.0: ok

Linux

Ubuntu 17.10

Sage 8.2: ok (requires sageRegina5.1.3 or later)
Older OS versions
In case the installation did not work...
Check whether the error you found is described on the troubleshooting page.
If not, please send an email to Matthias Goerner (enischte at gmail dot com) that includes the reginaInstall.log file and basic information about your system returned by the commands:
uname a
sage version
sage sh c "which gcc; gcc version"
sage sh c "which as; echo  as version"
If this is an issue with regina itself, rather the install script of sageRegina, please contact the original authors.
Where can I find the source code for sageRegina?
Here is the source code for the python packaging and for
regina.