I sadly I haven't read most of his stories yet(though I have most), but from what I can find, there are only mentions of "worldships" (planets as spaceships) and references to living stars, which I suppose could be considered a living ship, but not a bioship or organic ship.
Not sure about Old Olaf but isn't McCaffrey's The Ship Who Sang a bioship?
As far as worldships, I know both Heinlein and Blish wrote about them decades ago, Heinlein in Orphans of the Sky
, 1963 (fix-up novel comprising the novellas "Universe" and "Common Sense", both originally published in 1941) and Blish in his Cities In Flight stories.
Regarding the Singularity and computing limits in general-I believe I recently read an article about practical molecular computing, which would upscale current capabilities by 100-1000 times current speeds, although I can't find it now.
So who's to say that, or DNA computer coding might not lead to a Singularity effect?
I saw an earlier citation for a handheld computer. I know Clarke describes one pretty well in Imperial Earth-1975. And Norton describes efficient hovercraft in Star Rangers-1953.
Just some minor observations.