Would this be an example of promissory estoppel or a breach of oral contract? Or both?
For example, if a landlord orally contracts with an individual that the individual does not have to sign their lease to occupy their property for a period of 6 months so long as the individual pays the monthly rent, submits to a background check prior to move in, and provides renters insurance prior to move in, and then the individual does all these things at the landlord's request and gets the keys to move in-- can the landlord later then evict the individual halfway through the 6-month term claiming that the individual is an unauthorized occupant, or does their acceptance of the individual's money for the first 3 months constitute a promissory estoppel? Or is this a breach of oral contract? Or both?
A tenant without a lease agreement is a Tenant at Will, because regardless of the nonexistence of a signed lease agreement you occupied the premises with the permission of the landlord upon some form of "arrangement". As a tenant at will, you have the right to "lawful and exclusive possession" of the place you rent. And, in fact, you are entitled, in some jurisdictions, to a longer NOTICE PERIOD that most tenants with signed tenancy agreements. Please consult a UD (Unlawful Detainer) Attorney in your jurisdiction to properly advise you on the local applicable laws.
I said it could be both earlier, because a Promissory Estoppel would prevent the landlord from withdrawing the promise made to you when you have obviously reasonably relied on those promises. It appears the promises were unequivocal and there seem to be evidence that there is a change in position made by you as a result of the promise on which you relied, and eventually suffered some detriment, even though the law does not required that you necessarily have to have suffered any detriment. Moreover, it will be an inequity if the landlord were to go back on those promises - so, it appears all the ingredients and requirements of a promissory estoppel are there.
Also, there is a breach of contract as the oral agreement entered into by the landlord and you were entered into voluntarily with the intention of creating a legal obligation - and binding contracts can be made orally.
For the former, promissory estoppel, you may be compensated by mere reliance damages, which may also entitle you to the "benefit of the bargain" or specific performance.
For the latter, breach of contract, you may be compensated by being awarded damages of monetary compensation, and/or also specific performance, and/or an injunction, in equity.
Incidentally, Tenancy at Will is the most common type of tenancy, and contrary to what most landlords think and most tenants believe, Tenants at Will do have rights and cannot be "just booted" out of the premises. You are not an "unauthorized occupant" because you came in with the permission of the landlord - you just don't have a written lease - you may be entitled to an even longer Notice, if the landlord insists on evicting you, than a tenant who signed a lease.
Consult an Attorney immediately, you can go after your landlord legally to seek redress for these breaches. I wish you the best.