We cannot know for certain if something is an inferior good without actual data. We must also realize that some things can be normal goods for some people while, at the same time, they are inferior goods for others. That said, we can generally say that dry cleaning will likely be a normal good for most people.
The reason for this is that dry cleaning is relatively expensive and is only needed for clothes that are fairly special. People will most likely not buy the sorts of clothes that need to be dry cleaned when their incomes are dropping. If they already have them, they will be less likely to wear them because of the fact that they will need to pay to get them cleaned.
When people’s incomes drop, they are much more likely to opt for clothes that can be washed in a regular washing machine. Dry cleaning, therefore, is not a complete necessity. Since it is not a complete necessity and since it costs a relatively large amount of money, it is likely to be a normal good for the majority of people.