Alcohol is present in many deodorants and perfumes as a solvent or carrier. You can read the detailed post on the Fiqh of Ethanol.

When it comes to perfume, each ingredient has scent molecules of their own particular weight, for example, citrus oils have light molecules that are quite volatile and fade quickly, but heavier base oils like sandalwood and amber are heavier and don’t disperse much. In other words, you’ll only really smell them very close to the skin.

So the alcohol doesn’t only preserve, blend and stabilizes the perfume but it helps the scent molecules to disperse in the air around you.

Many antiperspirants contain alcohol, which is used to dissolve different ingredients and to help the product dry quickly when applied. The majority also will use isopropyl.

Alcohol in such products will generally be halal, 1. if it’s ethanol then the below will be considered, 2. if its methanol or another form of alcohol then such products do not intoxicate thus halal.

In summary, the ruling is based on the source of the alcohol not being from dates or grapes, else it would be impure conclusively. If from another substance or synthetic, then the following four conditions need to be met:

1. It is not being used to intoxicate
   
   Alcohol is not used to intoxicate.
   
2. Not being used as intoxicants are used
   
   Nor used to make the consumer drunk.
   
3. Not being used in an amount that intoxicates
   
   The amount is incredibly at low levels, less than 0.1%.
   
4. Not being used for vain purposes
   
   The alcohol is not added for a vain purpose in such products.