The latest screening being done for newborns born in the US is a pulse oximetry test to screen for Critical Congenital Heart Defects (CCHD). This screening is done between 24-48 hours postpartum as the baby's circulation makes the transition to life outside the womb. Motherland Midwifery offers this screening at our first postpartum visit. Depending on the nature of any abnormal results and the over assessment of your baby, we can consult with a pediatric cardiologist, or may refer your baby to your pediatrician or to the nearest pediatric hospital.
This blog post from Science & Sensibility provides a good overview of the process.
"Critical congenital heart defects refer to heart defects that babies are born with and that require surgical intervention within the first month (or year, depending on the defining organization). About 1 in 100 babies have heart defects (1%), and about 1 in 4 of those with a heart defect have a defect so severe that it needs to be corrected immediately (0.25% of all babies) Only some of these defects will be picked up by prenatal ultrasound, and they may not show up on exam before the baby goes home (or the midwife leaves in the case of a home birth). Depending on the defect, some babies may be able to compensate with structures that were in place during the fetal period but begin to go away after the baby is born."
This resource from March of Dimes is directed towards parents and has thorough and accurate information.