I guess I'm qualified to speak, as I adopted as a single woman -- actually long, long divorced -- back in 1997, and my daughter is now 21 years old. My daughter was 18.5 months old when I adopted her from China, and I was nearly 51 when she came home.
I can only say that adoption by singles is not only possible, but extremely common, these days. And singles do all kinds of adoption -- domestic newborn adoption, private adoption, domestic foster care adoption, and international adoption. I know many people, mostly women, who are single parents via adoption.
Nowadays, most agencies do not discriminate against single women. There are, however, some foreign countries that don't allow singles to adopt, and agencies must accept their requirements. And some countries and agencies may choose not to place with women who are openly lesbian. Single men DO face an extra level of scrutiny, because there is sitll an all-too-common belief that if a man wants to nurture a child, he "must be a pedophile" or "he must be gay" (said in a tone indicating disapproval of gay men.) Single men of any orientation may find it difficult to adopt internationally, or to adopt a newborn or infant, but if they are willing to jump through a lot of extra hoops, they can usually adopt from foster care or do a private adoption.
When I adopted my daughter from China, I was part of a group of eight families that traveled together -- five married couples and three single women, including me. We have all remained friendly, and my daughter was especially close to two of the kids adopted by single women and one adopted by a married couple. And we knew many other families headed by single women, many with children from China, but also children from other countries (including the U.S.), a child of a woman who had an unplanned pregnancy in the U.S. and chose to parent, lesbian couples, families headed by grandparents, and so on. One thing that tends to be helpful when you adopt as a single is living in an area where there are many different types of families, including families headed by single women. That way, your child doesn't need to feel "odd" for having only a Mommy. Most kids of single Moms go through a stage of wishing that they had a Dad, but they will go through it less traumatically if they see other families who don't fit the Mom-Dad-kid picture.
Another thing you should have is a strong social support network. Being a single parent is hard work, and sometimes, you will run into situations where you need an extra pair of hands to assist, a shoulder to cry on, or someone to provide respite because you have the flu and a temperature of 103. Your social support network can include helpful family members, but it can also include friends, and especially adoptive parents. I managed to shatter my kneecap when my daughter was two, and I really valued the help of my friends, old and new. Also, make sure that you have a support network that includes some decent male role models; both male and female children need role models of both genders. Again, these men can be relatives or friends -- even ex-boyfriends, if nice. I had planned for my child to get to know a wonderful male cousin and an old boyfriend of mine, who was the father of two and grandfather of three children. My daughter wound up liking them -- but her "main man" became the Japanese husband in a family that cared for Becca when I had to have surgery. He and Becca remain friends, in the nicest possible way, to this day, even though he has divorced and remarried; my daughter was a bridesmaid in the wedding, along with his daughter.
Single adoptive parenting can be very difficult; I don't want to mislead you. But I was surprised to find that it was actually easier than I expected, given all the gloom and doom scenarios that some people told me about. It IS true that you may never again have enough money, enough time, or enough sleep. But I can tell you that, at 71, when I look back at the wonderful life I've had, the most significant thing I feel that I've done was adopting my daughter. And the fact is, she's doing remarkably well. She's finishing college, with excellent grades. She already has a job for after graduation. She is self-supporting, except for the financial aid that her university gives her. She has had a steady boyfriend for over three years. She does her own taxes, reads leases like a pro, picks her own health plan, drives safely, flies to job interviews all over the country, and is hoping to get an MBA after working for a few years. Heck, she even keeps in touch with Mom!
My best wishes to you as you contemplate adopting. And don't hesitate to contact me if you want to talk by email or phone. If you'll PM me on this site, I'll send along my information.