* Detrimental to: This is the most common preposition used with "detrimental." It indicates that something is harmful or damaging to something else. For example: "Smoking is detrimental to your health."
* Detrimental for: This can be used when something is harmful to the overall situation or progress. For example: "The lack of funding was detrimental for the project."
* Detrimental on: This is less common but can be used when emphasizing the impact something has on something else. For example: "The decision had a detrimental impact on the economy."
Ultimately, the best preposition to use will depend on the specific sentence and the meaning you want to express.