I have downloaded Typescript plugin for Visual Studio, installed it and restarted my PC. I have also installed both VS 2013 Express for Web and VS 2013 Ultimate Preview but
According to the description here you cannot have the standalone version and the update containing the component at the same time.
I have VS2012 and VS2013 on my system and had a lot of issues with the typescript support (or lack of) with VS2013.
So I uninstalled the typescript 1.0.0 tools for Visual Studio 2012 and re-applied Visual Studio 2013 Update 3. It is now working fine but some details, the project migration were not done properly, I had to exclude the ts and js files, re-include the ts file and make sure that the build action is "TypeScript compile".