How do I install OS X Yosemite without a working Mac App Store application? My installation of the Mac App Store on Mavericks is broken. What alternatives are there for installing the new OS?
My MAS application is still broken and there are glitches everywhere.


Looks like FUBAR BUNDY!

I recommend a clean install.


Backup any additional files which aren't already in your SuperDuper! image. Probably it's faster to just backup/copy your or all user folder(s) to an external drive.
Please check your hardware with your original system DVD (Apple Support: de_DE en_US) or if the DVD is not available with the appropriate Apple Hardware Test image.
Do a SMC-reset (Apple Support: de-DE en-US)
Do a PRAM-reset (Apple Support: de_DE en_US)
Since your recovery volume is probably corrupted, you have to initiate an internet recovery by restarting your mac and pressing alt cmd R.
The prerequisites for an internet recovery are listed here: supported Pre Lion Macs and Recovery/Internet Recovery.
In a few words you need the latest firmware update installed, either ethernet or WLAN (WPA/WPA2) and a router with DHCP activated for a successful internet recovery.
On a 50 Mbps-line it takes about 4 min (presenting a small animated globe) to boot into the recovery netboot image which is loaded from an apple server.
(Additionally you might need an external drive or a thumb drive with at least 32 GB if you want to install to an external disk.)
In the opening window choose Disk Utility and format/partition the drive you want to install the system to. Choose Mac OS Extended (Journaled) and GUID partition table (available with the option button). Quit Disk Utility.
Choose Reinstall OS X and Continue. After some verification steps you will be asked to choose a partition for the new system. Continue until the system is installed. Depending on your internet connection it might take several hours to download and install 5.5 GB. If you get a message like 'Could not find installation information for this machine' just restart your Mac and continue with step 5. (The netboot image seems to be very picky with the WLAN connection and it took me two attempts to get around this step)
Set up your new system.
Start App Store.app and log into your Apple Store account.
Download and install Yosemite if it wasn't already installed in step 7 or just download it and quit the installer to get the Yosemite installation app for any further use.