Soft, and not rubbery (angel food cake is what I think of as rubbery). Very moist, poind cake is about perfect, and not with the traditional buttercream frosting. I find buttercream frosting to be way too sweet, and stiff. I also don’t like the taste or texture of fondant icing.
I’ve had some type of whipped cream types of frosting that were velvety smooth, and not too sweet. I’ve also had another kind of frosting that was also very smooth, and not so sweet. I think it might be called flour frosting. I also like powdered sugar icing type of frosting, like the kind you drizzle over gingerbread cake and other quick breads.
I also don’t like cakes that are too sweet, such as most chocolate cake, although I do like German chocolate cake which is a more mild taste to me. But I actually prefer fruit flavored cakes the best, especially lemon, and strawberry, and I like the filling to be some type of jam, or whipped cream, but not buttercream or custard.
If given a choice, I like cake served cold, rather than at room temperature. Somewhat dense (as opposed to angel food cake, which has too much air).
Baskin Robbins used to make excellent ice cream cakes. Don’t know of they still do.