My own personal solution to this has become fairly dependent on two tools - one hardware, one software:
Hardware: Fujitsu ScanSnap Scanner. I have the S500, which is now something like 5 years old, and is still running great. This thing beats the pants off of scanners that are built into typical "all in one" print/copy/fax/scan devices. It rarely jams or sucks in multiple pieces of paper. And it scans both sides of the paper in one go, which speeds things up quite a lot. I
Software: Evernote. Running on my Windows PC, Evernote has a neat function where it can import in all scans it finds in a particular directory. And then it does its OCR magic, which makes most documents (even my handwritten ones) text-searchable.
So the combination of these tools means that I can take handwritten notes wherever that works, and just scan them into my computer later, via the ScanSnap. So my goal is to either take notes in Evernote directly, or to scan handwritten ones into there later. Evernote easily syncs across multiple devices (for instance, laptop-desktop-smartphone), so this encourages me to make sure all my notes are eventually corralled into one spot - my group of Evernote notebooks. They are also accessible via the web, so basically anytime I can reach a computer, I can reach (and search) my Big Stack Of Notes.
Eventually I just shred or throw away the paper version of my notes.