I wish I had a full answer, but I had asked this very same question before, and it doesn't show up in the question list when I'm not signed in, so I don't know if it was just 'dismissed', or what.
What I DO know, is that when I run the appcenter in terminal, then I type two characters into the search bar, the app crashes because of a "Segmentation fault"
[INFO 18:53:31.544223] Application.vala:155: App Center version: 0.1.2
[INFO 18:53:31.544336] Application.vala:157: Kernel version: 4.4.0-47-generic
This is what happens when I input two characters:
[FATAL 18:53:40.229094] Client.vala:241: The package id's '' are not valid
Segmentation fault (core dumped)
What I also found is that if I type in 3 or more characters before it has the chance to crash, it works just fine. What this tells me is that it isn't the search function that has the bug, but rather the handling of the first two characters that are put into the search bar.
At this point, I hope a developer comes across this post and knows exactly where to look from here.