It is far, far better to be single than to be in an unhappy or unfulfilling relationship. Our culture puts so much pressure on people (particularly women) to pair up that few people stop to think about why that is.
What's actually wrong with being single? Loneliness? Quite a lot you can do about that even as a single person. Poverty? Why should you be poor when you as the parent with care are the one person whose been responsible and is living up to her obligations? Maybe it's society that should change, not your single status.
There's quite a lot that's right with being single.
So you have to do it all. Unless you have enlightened partner who respects women and believes in domestic equality as well as equality in the workplace, you'll find that even in a relationship you still have to do it all and pick up after another adult and worry about whether they are happy. At least on your own the only mess is your own and when you do finish doing it all you have some genuine 'me time'.
And if nationwide stats are anything to go by, 2/3rds of men still don't pull anywhere near their weight at home. I suspect the wives/partners of the 1/3rd that do are the ones who really do have it all, meaning most don't. Remember that nearly 2 in 3 relationships fail, so all those wonderful relationships you think you're seeing are definitely a facade in many cases.
If you have good friends and the ability to do something that you find personally fulfilling, you may just find that not only do you not mind being single, but you actually quite enjoy it. Of course, money plays a big part (much easier to find a fulfilling hobby and maintain friendships if you have money), so for anyone on a low income it is doubly hard. Lone parents are 2x more likely to be living in poverty than any other sector of society. This is why the recent changes to the CSA are a travesty, why more needs to be done to get maintenance considered as fundamentally necessary as paying tax, and why childcare needs to be much more heavily subsidised. But that's a whole other issue.
It's natural for humans to want to bond with others and to want sex, but that doesn't necessarily equate to life-long marriage despite what the Tories, bible and Daily Mail would have us believe. There are all sorts of ways to achieve those goals that are neither immoral nor damaging to children; it's simply a case of finding out what works for you.
Conversely, of course, once you have a life you love as a single person, you are far more likely to find the right sort of person for a happy, respectful long-term relationship.