In this case, file the claim immediately. And do just what the above poster suggests. DON'T talk too much about what you've heard. Just repeat exactly what you were told by your employer on the last day you worked. If you have a termination letter or something that you were given, turn that over. But do not go into speculation, office gossip, or what you believe may have happened or what you think might be the reason you were terminated when applying for unemployment. Just say what they told you. Let them tell the unemployment system what they are going to.
It will take several weeks. They will take what you say, and take what the employer says when they call and speak with them. They may, at some point, call you back to seek clarification of the issue. If your employer chooses not to respond to their inquiries, or says you were terminated because of downsizing on the job, which would be the right thing for them to do, of course, (and would certainly make it harder to show age related bias, which I'll talk about later) you'll probably be approved right then, initial decision, nothing else necessary. Eventually, making your weekly certifications for benefit weeks as they pass, you'll start drawing a claim. That's the reason you file immediately. You can only be back paid for weeks that have passed since the claim was filed, even if you are approved.
If they are stupid, and start down the "Well, she did something wrong!" road, DO NOT fall into self justification, repeat that office gossip, etc. Stick to what you told them initially. You did your job to the best of your abilities, to your knowledge, you were following every rule and procedure correctly, and you were told you were being terminated, that's all you know. If the adjudicator calls you, and says "Your employer says blah blah blah...." you still stay with your story, which happens to be the truth. The employer never told you anything, you never received any warnings or write ups, you did your job to the best of your abilities. To your knowledge you were an excellent employee. You were terminated.
In order to keep you from being approved to draw benefits (which cost the company money) the employer must show that they had a valid, work related misconduct reason to terminate you. If you unknowingly did something, approved something you shouldn't have, that's not misconduct. In this case, it will come down to whether or not they want to take this whole thing further, argue that your reason for termination was some actual misconduct, or whether they'll just let this one go. They don't seem to have any proof or documentation of your having done anything deliberately wrong. Stick to your simple story and see how it plays out.
Now, after you have filed for unemployment insurance benefits, you may want to think about the EEOC. I'd possibly do a consult with a labor attorney (free of course) and let them tell you if it appears you might have a valid case for age discrimination. If so, this would be a long, drawn out process, begun by your filing a claim with the EEOC, and you would probably be long through with your unemployment benefits and reemployed before any case came to fruition IF there is determined to be any case.
But you'll find out a lot about what the employer is trying to pull off with the unemployment insurance claim, though it is NOT related to a discrimination complaint and no information from the unemployment issue will be passed on to any other agency or something like that. It would be up to the EEOC to determine if the employer in your case, and possibly in other cases, targeted you because of your age (iin other words, for an illegal reason.)