I don't know of any research. But I'll throw in some opinion anyway........just for the fun of it.
Assuming we are talking about washing machines and fridges, it would seem to me that kitchen equipment that uses heat might be the more common source of fires - the cooker, chip pan and microwave. But anyway.....
I would speculate that the fridge and washing machine tend to have flat, sealed metal tops and that we could not expect sprinkers to spray water into these, in the early stages of the fire they are going to provide some cooling and perhaps stop/delay the fire spreading to nearby combustible items.
The loss of the fridge, plus the associated water damage would for sure be a better outcome than the fire spreading to the whole kitchen or whole house and the inability of sprinklers to penetrate domestic electrical items like this isn't something I'd be worrying about, I'd be more worried about the bigger issue of the complete destruction of the property the life safety risk.
......while acknowledging that this isn't what you asked......