Depends on the adoption type and where you live.
I adopted as a single parent through foster care, cost me $200 for a new birth certificate for her. I got married a few years later and my husband adopted her too, again $200 for a new birth certificate to add him. We also adopted a little boy together through foster care. Adopting through foster care is very minimal if any costs but often children available for adoption are over 3 years old. My daughter I've had since she was a newborn, my son was 4 years old when he was placed with us.
Other adoption types like direct placement can cost a few thousand, mostly legal fees. Direct placement is when you know the birth parents or are connected to them directly by someone. Domestic infant/newborn adoptions can be $20,000+.
All adoptions require a home study. A home study consists of a criminal background check, fingerprinting, a financial disclosure, 3-5 references, interviews with you, a physical from a physician, parenting classes and a physical inspection of the home.
Most domestic infant adoptions the birth parents choose the adoptive parents and being single can be a disadvantage. Many want to place their baby in a 2 parent home. However, some can and do choose a single parent.