I don't like any of the gear designs and the "signed first patent" designs, um, I think that would be numbers 4-8.
I do like the one with C code (number 9), but only for the hilariousness - there's no way it would pass!
3 is just generally ugly. 10 is weird, while not being funny (though I'm probably just not getting the joke). Ditto 11, to a lesser extent.
12 is awesome but won't work well on a coin; same for 13, sadly.
Is the picture on 14 the thing the first patent was for? If yes, I like it. If no - same category as 11.
(It looks like it might be, but I can't figure out enough to be sure either way.)
...That leaves the simplicity of 1 and 2, which I don't have much of a preference between. 2 is more coin-like; 1 is prettier.
As for the obverses... well, to start with, 1 and 11 are just outright ugly.
So is 9, come to think of it... Lady Liberty isn't supposed to look like Donald Trump. Wonder if that was deliberate.
Not a fan of 10 either; the flag doesn't really have a place here, and the book is also weird.
I don't like how 2, 3, and to a lesser extent 8 have the motto as the main part of the design, while not even positioning it properly; 4, 5, and 6 (and 9, if we ignore the Trump similarity, and to a lesser extent 12) also feature a huge motto, but at least have the decency to position it nicely.
That leaves 7 as my favorite (none of the problems), closely followed by 12 (only mild problems, and much prettier).
Not sure what would take the third favorite position, though. I like how 6 is reminiscent of the Panama-Pacific gold dollar, but those days nobody is going to use (and almost nobody is going to collect) a coin the size of a gold dollar, and it's not going to work in larger sizes. That motto...
What I would have changed for a perfect favorite: 8, but with the motto along the rim on the right (and the denomination enlarged on the left).
Random fun fact: Samuel Hopkins, who got the first US patent in 1790, apparently also shared the first Canadian patent, from 1791, with Angus MacDonnel, then of Quebec City.