I've been to the Sultan's armouries at Topkapi Palace in Istanbul, and that seemed like a reasonable place to start looking. It's not hard to find recent pictures online claiming to be of the Sword of Osman (Wikipedia has one dated to 2014, but says nothing about where it was taken) so it seems to be in a museum collection. But I couldn't find anything about it through searches related to the Topkapi Armouries.

After a bit more sleuthing through Turkish language sites, it looks like my initial guess was close. But rather than store and exhibit the sword in the armouries, it appears to be part of the Ottoman Treasury collection at Topkapi (Fourth Hall of the Treasury, if Google Translate is not leading me astray), which I suppose makes sense since it is something more historically significant than a military sidearm.

But this page makes an interesting point, that the Sword of Osman might properly be considered even more than a treasure, and perhaps even a holy relic, in which case it should properly belong in the Holy Relics collection. But the write-up suggests that the provenance of the sword might be in question, so it has been downgraded to mere treasure. Google's translation is bit too clumsy to work out much more than that, and my Turkish does not go much beyond kılıç.

This is not terribly authoritative, I'm afraid. But that's where I would start my search: Topkapi Palace Treasury, Fourth Hall.

I was in the Topkapi Museum two weeks ago. I was in the armoury, the treasury and the sacred relics departments. I didn't see the sword, and I was looking for it specifically. Perhaps I missed it in the treasury collection however. Someone should please say if they saw it.