Breast Cancer Screening Safety Act of 1991 - Amends the Public Health Service Act to require certification in order for a facility to perform or interpret mammograms, inspect equipment, perform needle localizations or other procedures using mammography equipment, or conduct quality assurances oversight related to mammography. Authorizes the Secretary of Health and Human Services to issue and renew certificates for up to two years.

Allows the Secretary to approve a private nonprofit organization to be an accreditation body which meets certain requirements, including inspecting facilities.

Directs the Secretary to: (1) establish standards for facilities to assure the safety and accuracy of mammography; (2) specify organizations eligible to certify individuals to perform radiological procedures; (3) specify boards eligible to certify individuals to interpret screening mammograms; (4) establish standards regarding the qualifications for individuals to interpret screening mammograms; and (5) specify boards eligible to certify individuals to inspect screening mammography equipment and oversee quality assurance practices.

Directs the Secretary to conduct annual announced and unannounced inspections of certified facilities.

Provides for: (1) intermediate sanctions for certain violations; (2) suspension, revocation, and limitation of certificates; and (3) injunctions. Requires fees for certificate issuance and renewal, and for inspections, sufficient to cover costs.

Requires annual publication of a list of facilities convicted of fraud and abuse, false billings, or kickbacks, facilities that have had certificates revoked, suspended, or limited, and facilities that have been the subject of a sanction or other similar matters.

Allows the Secretary to exempt facilities in a State with more stringent requirements from compliance with this Act.

Directs the Secretary to: (1) make grants for research on new methods of establishing a Mammography Registry; and (2) based on the research, establish the Registry.

Amends title XVIII (Medicare) of the Social Security Act to require that screening mammographies paid for under Medicare be performed by a facility: (1) certified under this Act; and (2) in compliance with specified provisions of this Act.