The suggestions for other activities where parents are present are key. I am only just now starting to sort of know the other moms at my daughter's daycare after a year and about 8 birthday parties (which at daycare level, all the kids are invited, and the parents have to come too, so it's different than for your daughter's age). But even though I now know their names and who's kid belongs to who, we still don't get together or anything. A couple of moms have tossed out the "playdate" suggestion to me, and I haven't made it a point or followed through-neither have they-just because we're all swamped.
But also, if it's a parent who I know has a spouse and a busy schedule who's asking, I don't sweat it, but if it was a single mom, I would make a point to meet up, because I am sensitive to how hard it is after being single a very long time before getting married myself. You feel like everyone else is too busy for you.
Actually, most of MY friends where we live, don't have kids!!! I met them when I first moved here about 5-6 years ago before I had my first and was out more. And many are single. These people are naturally less busy and available more often to talk, meet for a drink, etc. And between caring for my own kids, (my husband travels constantly) doing kid activities and events and RARELY meeting up with a friend or two of my own-that's it for my social calendar. My closest friends from the past are out of state and now phone calls and emails-which also take time.
One key factor that really helped me, is that one friend I met here is extremely social naturally. She networks like a pro and makes a point of attending and hosting lots of "things", so as long as I have a link to her, I can always reach out and find something to attend, or have her help me do something where people are invited, because I too would be that person whose invites went unanswered. Socializing is a gift and an art not everyone has.
But as for the friends I have here (including her), I had to go against my hermit nature and reach out to them, invite them out several times, and accept invites to group nights out that I wasn't in the mood for. It does take work. I think you should find groups in your area you can relate to regardless of parent status, as WELL as do things with your daughter that include parents. If you and your daughter are BOTH socializing, it helps. My social friend has no kids, but has introduced me to several other nice parents.
So, if I were you, I would
1) Look for social activities for yourself and keep an eye out for "social ring leaders"
2) Find activities with your daughter that INCLUDE parents on a repetitive basis. Once you notice one or two moms who seem like people you could get along with, make a move. Don't be afraid to tell people you're new to the area and a single mom and you need friends. If someone told me that, I'd be sure to reach out more.
Be persistent and don't feel bad if it takes time or a few "no answers" before you succeed. You really are new there, even though it seems like so many months. It takes a long time, we don't have the neighborly chivalrous codes today that communities used to have.
Tell your daughter to hang in there too. I had no overnights or parties at that age either. My parents both worked full time and had almost no friends. I always had one or two close friends, not bunches, and no really close friends until junior high-ish.
You know what I recently did? I latched on to two moms having a joint birthday party for their daughters. I did it because my husband was out of town, I was overwhelmed, and I wasn't going to be able to do a party on my own. I went up to the daycare owner and asked for a class list of who had birthdays near my daughter. They were already close friends of each others, with their daughters in gymnastics together as well as daycare, I didn't know them. I horned on in, begged, offered to go in on the costs etc. At first, in my own mind, they seemed annoyed and distant and slow to answer my emails and questions, but I didn't care, I just persevered, offered to do most of the work, stayed chipper. By the end of the 2 week planning ordeal and the party that was way better with three people's input, we all get along and know each other. Don't be afraid to approach people who are already friends and be pushy. You know you and your daughter are nice, they will learn too!