Custom shapes can be added as a path(via the options at the top of the application or use the shape layer option), use the 'export paths to ai' command found in the file menu to save the path as AI format. You can then use the resulting vector file in many other applications. Once you have opened the exported the artworks, the generated AI file has no color fill or stroke, you can add that in many apps such as Illustrator.
1) Select the custom shape tool
2) Click the path option (top bar of the app)
3) File menu and Export command and Paths to AI
They can also be copied to some applications via the pasteboard / clipboard
A quick way to export the CSH presets to Illustrator and other apps is to use the recently added CC libraries (found via the window menu). Select the layer and drag to the CC library. You can then go to Illustrator and drag the artwork to the artboard and select the first option (convert layers to objects) to keep the artwork editable. You will have to release the clipping mask etc and remove any bounding boxes (nothing is ever made easy) and then you can edit the artwork such as moving anchor points etc. See the same path being edited in AI below
You can also export the artwork as a SVG file. Create the paths in PS and then go to the file / export menu and select the export as and select the SVG format. The SVG format can then be imported into AI as well as Dreamweaver (as used throughout this site with many a vector path displayed in SVG). You can also re-open the SVG file in PS though sadly it is not treated as a vector at that point (but it is in Illustrator)
You can also export them via the save command by selecting a format (blend with the background or use as normal raster layers) such as PSD or PNG or TIFF format. You can, if you wish to spend the time, go through all 13,000+ shapes and create huge 2000 x 2000 files and save each and everyone of the designs to PNG. Just an idea for a wet Sunday afternoon. The artworks can be exported for use in PSP, Adobe Indesignesign, Affinity, GIMP etc via the save command. There is an app called PresetBreeze from Tumasoft that export the artworks from the CSH, that may speed things up a little.
Probably the best all purpose format is PSD, the native Photoshop format. You can save raster and vector data with that format. Just use the save as command