Yes there are a couple of search modifiers you can use to narrow down your results with regards to accepted answers.

hasaccepted

hasaccepted is used for when you want to know if there is an accepted answer on the question.

• hasaccepted:yes: The question has an accepted answer

• hasaccepted:no: The question has no accepted answer

isaccepted

isaccepted is used for when you want to find answers that are, or aren't, accepted.

• isaccepted:yes: The answer is accepted

• isaccepted:no: The answer is not accepted

Putting this together and taking your comment, "these are questions with one or more answers but they've never been ticked as 'answer accepted' by the OP", into account I imagine the query you are looking for is this, where answers specifies the minimum number of answers on the post:

hasaccepted:no answers:1

Then taking into account you originally tagged this meta post with story-identification you are probably looking for this query:

hasaccepted:no answers:1 [story-identification]

Yes there are a couple of search modifiers you can use to narrow down your results with regards to accepted answers.

hasaccepted

hasaccepted is used for when you want to know if there is an accepted answer on the question.

• hasaccepted:yes: The question has an accepted answer

• hasaccepted:no: The question has no accepted answer

isaccepted

isaccepted is used for when you want to find answers that are, or aren't, accepted.

• isaccepted:yes: The answer is accepted

• isaccepted:no: The answer is not accepted

Putting this together and taking your comment, "these are questions with one or more answers but they've never been ticked as 'answer accepted' by the OP", into account I imagine the query you are looking for is this, where answers specifies the minimum number of answers on the post:

hasaccepted:no answers:1

Then taking into account you originally tagged this meta post with story-identification you are probably looking for this query:

hasaccepted:no answers:1 [story-identification]

Note that this information is easily found on any "search results" page, there is a hyperlink at the top reading "Advanced Search Tips" and when clicked expands to show some search modifiers.

At the bottom of this page is the line:

For more details on advanced search visit our help page

Which links you to a more thorough help page on all the modifiers if you want some more in depth information on them.