Are there alternatives to mammogram for breast cancer screening?

None as good. Mammograms are the best test for detecting breast cancer. It is not clear that breast exams by a health care provider add benefit (tho they may), and breast self exams have not been shown to be beneficial. Certainly, if you notice a bump, a change in skin, a nipple discharge or any abnormality that you are concerned about, you should get it checked out.
Mammography. is currently the standard of care for screening. It has the most scientific validation and, whatever its imperfections(no test is perfect), all of the other potential screening modalities are more imperfect for various reasons. Ultrasound and MRI have niche screening roles in special situations.