Formal:
* J'ai sommeil. (Most common and neutral)
Informal:
* J'ai envie de dormir. (I want to sleep)
* Je suis fatigué(e). (I am tired)
* Je suis crevé(e). (I am exhausted, literally "I am burst")
* J'ai les yeux qui piquent. (My eyes are stinging)
The best way to say it depends on the situation. If you want to be polite, use "J'ai sommeil." If you want to express a strong desire to sleep, use "J'ai envie de dormir." If you are just feeling tired, use "Je suis fatigué(e)" or "Je suis crevé(e)."
Let me know if you have any other questions!