Polynya has a great post of classifying rollups in his tweet here : https://twitter.com/poIynya/status/1511623759786307586
Copying here for ease.
Regular rollups (I have called them discrete rollups or secured rollups in the past) - Arbitrum, dYdX etc. These rollups are chains where instead of a consensus, security is coordinated through smart contracts on a highly secure and decentralized chain instead, e.g. Ethereum. This way, they get all the benefits of a centralized L1, and then some, but inherit the security of Ethereum. With one caveat - these rollups are upgradeable through the rollup’s governance.
Immutable rollup - so, the next step is a rollup whose smart contracts aren’t upgradeable (e.g. Fuel V1). There are no external governance risks, but the downside is upgrading requires going through the EIP process, or deploying a new instance - like we see with DeFi protocols
Enshrined rollup - going further, don’t have any smart contracts or governance risks at all - simply have the rollup in-protocol. It’s no longer an “L2” but rather getting all the immense technical benefits of rollups at the L1 level. The downside is upgrades can be very slow.
Sovereign rollups - these rollups settle themselves, instead of a more secure settlement layer. Instead, they just use the DA layer purely for data and ordering. These do not have secured bridges to a settlement layer, which IMO misses the biggest feature of rollups.
Bonus round: honest minority sidechains with rollup fallback, e.g. Arbitrum AnyTrust. These chains operate basically like any alt-L1 or sidechain, but with significantly higher security due to 1-of-N consensus, rather than 67%-of-N. The trick is if even one (or two) validator dissents, the chain falls back into a rollup. As such, AnyTrust chains are strictly superior to alt-L1s, and I’d argue also to sovereign rollups. Of course, secured Ethereum rollups are the gold standard, and will become negligibly cheap EIP-4844, and danksharding.
By the way, an execution layer which settles proofs on one external chain, but data on an different external chain, is not a rollup. If validity proven, such a construction is called a validium.
As I understand, it is easier to deploy rollups on MINA as the circuit currently in protocol allows for a zkp validation. Thus any rollup spawned can be depolyed as zkApp with its transition proof posted to the smart contract. Each of these rollups - could have its own consensus/upgrade mechanism as long as the zkp holds true. These rollups would still need to have a bridge - of sorts built into the zkApp to move assets across. DA layer is a bit of a mystery to me as there is no on-chain storage in MINA. This leads me to think atleast gen 1 rollups on MINA are going to be flavour of validium.
I would also like to open discussion on should the core protocol be enhanced to include enshrined rollups. I.e pretty much formalise a rollup specification thus by-passing a lot of the ground work at the cost of tighter coupling.