I use engineering paper and a mechanical pencil extensively for software development work, a habit I picked up in my chemical engineering days.
None of it is saved. I use it to extend my working memory for complex design problems, and for this purpose I find it very effective. The advantage of paper and pencil is that it is naturally and efficiently amenable to non-linear and somewhat arbitrary patterns of access and representation.
Software really struggles at this because computer UIs force everything into coarse linear workflows no matter the presentation, which makes navigating and rewiring a large number of orthogonal dimensions awkward and inefficient on the best days.
As we're absorbing this idea... I've seen employment contracts for technical people that arguably might consider these handwritten notes to be IP artifacts, which must be preserved and made accessible to the company.
If that's your situation, you could scribble the date, your name, project, etc. in some margin of the piece of paper, and stick it in a pile/folder. To eventually be scanned, and the file put somewhere accessible, and/or the paper technically preserved.
In a startup, you might also want to preserve things like these for nostalgia, even if they have no IP value. One seed startup, I kept a folder of "nostalgia" photos, video clips, screenshots, etc., of early prototypes, people working late, incremental successes, etc., and it was quickly a good thing. It's amazing how well you can progress, and what positive feelings you have some of these things that quickly become nostalgic. Anyone can see the current state of the code in the repo, but here's the pencil sketch on quadrille paper (or a napkin) one evening that became our architecture.
At least in my case, it is difficult to overstate how unlike linear “notes” those scribblings actually are. They aren’t remotely consumable outside the context in which they were created. It really is treated like swap space for working memory.
I do go through all the old papers when I discard them but 99% of the time there is no discernible value. The thought process that created them is strongly non-linear, which does not lend itself to naive linear consumption at all later date.
None of it is saved. I use it to extend my working memory for complex design problems, and for this purpose I find it very effective. The advantage of paper and pencil is that it is naturally and efficiently amenable to non-linear and somewhat arbitrary patterns of access and representation.
Software really struggles at this because computer UIs force everything into coarse linear workflows no matter the presentation, which makes navigating and rewiring a large number of orthogonal dimensions awkward and inefficient on the best days.