This question is similar to another one on here, but the details of my situation are not quite the same: Affiliation in the transition period between studies at different universities
I graduated college recently and applied to and was accepted to grad school. However, I deferred my admission for a year so I could complete a fellowship, which I have not yet begun.
1. How should I describe my academic affiliation now?
2. How should I describe my academic affiliation once I start my fellowship?
3. How should I describe my academic affiliation once I complete my fellowship but before I begin grad school?
It seems improper to list my former university, since I graduated and therefore should now be completely unaffiliated from them. Also an undergraduate has a tenuous academic affiliation to a department at best, at least in my opinion. On the other hand, the fellowship isn't really an affiliation with an institution either (even though I formally had to register at a university as part of the process, although I actually haven't completed the registration yet since this needs to be done in person). And for the grad school I haven't begun studies, and even though I was accepted and have an e-mail address there now, I feel like I would be counting my eggs before they hatched.
Honestly I know my situation is kind of strange, and maybe it would be best to say 'unaffiliated', but at the same time, for example if I posted something on ArXiv, I wouldn't want people refusing to take it seriously at all because I am 'unaffiliated', especially if they wouldn't think that way if they knew the specifics of my situation. That being said, they lack a snappy quick description.
Despite the differences in circumstances, I believe that the top answer in the linked question can be adapted to your case as well. Now, during the fellowship, and during graduate school, you will be registered with some university.
The fellowship period is a little odd, but they had you register with a university for a reason, and it seems that you will be affiliated at least tangentially---unless people with your fellowship list themselves as an affiliate of the institution giving the fellowship, which happens in some cases as well. Check and see what other people with your fellowship have done, and it should be clear.
During the gaps, use your last affiliation on publications, since you will almost certainly have done at least some of the work there. As for describing your current situation during the gap, you don't have to say "unaffiliated", you can just be clear, e.g., "I finished at X in June, and am starting at Y in September."