Not many as most are only open to residents and citizens of the US. You will have to do allot of searching and read very close the requirements for the scholarship. A part of the reason as to why it it is difficult is you would be a non-resident on a student visa you would not be eligible for any scholarship offered by any State or US government as the condition of the student visa is no support from the government, this means State or Federal. Another reason why you many scholarships are not available to non-residents is due to your student visa states you are to have enough funds to support yourself and this includes school. If you get a scholarship the implication is that you do not have enough to support yourself and you need to look for other means to pay for your school.
On an different perspective regarding your question, if you are in the country illegally it would invalidate any award due to the fact you do not have a legal right to be in the country. Furthermore it might also open you up the possibility of being charged with fraud or theft by deception in addition to the laws you violated by entering the country illegally. Since you do not have a legal right to be in the country you denied someone who had the right to be in this country legally from getting it, you deceived those who offered it, and in essence lied.
