How to search the debug console in vscode?

Since August 2020 and VSCode 1.49, the Debug Console does support filtering:

Debug Console filter

The Debug Console now supports filtering, making it easier for users to find the output they are looking for, or to hide irrelevant logging output.

The filter also supports exclude patterns (for example, patterns starting with an exclamation mark !).
The filter only applies to program output but not to evaluations run by a user.

Like other input boxes in VS Code, you can use the up and down arrow keys to navigate between old filter inputs.

In the short video below, the output is filtered to entries that include the text 'http', do not include the text 'http' (using the filter !http), and then include the word 'Severity':

^{(Click on the picture to enlarge. Look for the filter at the top right section of the animation)}

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And with the upcoming Sept. 2020 VSCode 1.50, you know where you are in your search:

Since last milestone the Debug Console supports filtering, making it easier for users to find the output they are looking for, or to hide irrelevant logging output.

Now we have added a badge to make it clear how many items are filtered away.

Looks like this answer is now outdated, see the accepted one ☝️

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A type-to-filter functionality has been added. To use it:

1. Give focus to the debug console (By clicking it, for instance.)
2. Press cmd + F / ctrl + F
3. Start typing

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).