This is currently a feature request, so I know the answer isn\'t straight forward, but I assume some clever people found clever workarounds for this.
One way is to copy/
Looks like this answer is now outdated, see the accepted one ☝️
A type-to-filter functionality has been added. To use it:
cmd + F
/ ctrl + F
You will see what you type in the upper right corner of the debug console.
You can then "enable filter on type" (the 3 lines left of the 'x') and only matching lines will show in the debug console. Press Escape to clear the search.
So it's not exactly the search feature OP was looking for, plus you can't type spaces, but it's the best built in tool for now (July 2019).