Can anyone help me identify what type of aloe these plants are? (they're all shoots from the same plant)

My guess would be Aloe aristata. Why is the exact species id important to you?

Looks like Aloe aristata or a hybrid between that plant and a Gasteria. A useful identifying feature for aristata (other than the patterned leaves) is the terminal bristle which appears at the end of the leaves. The Gasteria hybrids tend to have wider leaves and less of this bristle. Look at some pictures and see what you think.

Knowing if the plant is aristata might matter in an outdoor situation given that species is unusually cold tolerant, but indoors the exact ID does not really affect how you would care for the plant.

My first thought was Lace Aloe (Aristaloe aristata) too but I didn't think about Aloe/Gasteria cross, like Gasteraloe (XGasteraloe) which looks similar.

Thanks all - aloe aristata looks likely from a look at pictures. Part of what prompted me to ask (though I've been curious for a while) was that the largest one has started to flower - hopefully that will help to confirm the identification once fully underway.

Because botanical names are based on the reproductive characteristics of plants, plant flowers are indeed the best way to get a definitive id.