Can I filter the backlog?

Yes, you can filter the backlog, and you can create saved filters. To get started, select "Filtered items" in the "Show" dropdown in the lower right corner of the Backlog page:

Then add filter criteria like this:

You can combine filter criteria using the AND, OR, and NOT criteria:

For example, click "OR" to add an OR criterion, then add the criteria that should be inside the OR criterion. When you add a criterion, and there is a currently selected logic criterion, the new criterion will be placed inside the selected logic criterion. Note that you cannot place multiple criteria directly inside a NOT criterion. Instead, use an AND or OR criterion inside the NOT criterion.

The filters you create are private to you, and belong to the specific project that you create them in.

Note that, by default, completed backlog items are left out of the result when you filter. That's because you're typically not interested in the completed items, and it would clutter up the result if they were always included. To get them included, add a Status filter criterion where you include the statuses that you are interested in (for example, "Done" and "Sprint completed"):

You can hide the backlog items that don't match a filter by checking the "Hide other items" checkbox in the filter. The checkbox will appear when you've added the first filter criterion to the filter:

You can save a filter by clicking "Save this filter:"

The filter will then appear as an option in the "Show" dropdown.