commentator
No background check is going to bring up unemployment information because this information is not available to the public for background checks. Information about unemployment claims can only be provided to a third party under the strictest of restrictions, such as submission of a power of attorney. However, I strongly advise that any application you fill out, not necessarily your resume, but a company application that you are signing, contain all the places you have worked. They are not going to have any sort of access to whether you filed for benefits, were approved or denied, or what happened on any unemployment appeal unless YOU give them this information.
Whether you filed for unemployment after your short term job, and were approved or denied is not a source of shame. It certainly should not be a reason why an employer would fail to hire you, as of course in case of a layoff or an unjust work situation or termination, you would exercise your right to file for benefits. In fact, I'd simply say that the new position did not work out, not bother to mention that you filed for benefits, or whether the claim was approved or denied.
I can tell hundreds of "busted" stories. If incorrect information is on the company application, and signed and certified to as factual by the applicant, then you can be terminated at any time the company finds out you falsified the information. Most ordinary way people are busted? Other people tell on them. The work friend you confide in, the jealous ex-girlfriend, if you've got a lie hanging out there, all it takes is one phone call to get you in trouble. I had one jealous "friend" send us print-outs of a person's Facebook pages in which she bragged about working under the table while receiving benefits. We could not provide any information to the jealous friend about what was done or even whether the person was receiving benefits, but you can bet it was checked out.