Been a single parent throughout. Toughest bit is how it's all on you, the finances, the emotions, the decision making, the lot. It's also the most liberating bit. No fitting in a partner's hobbies, work events etc. No disagreements about how to raise them. But also, noone to share the joy with. Someone to share the firsts with. Childcare costs are a bitch at the best of times, but worse when your salary has to cover both that and all household bills etc. UC will cover a lot of it, so find out what you're entitled to. Do not budget as though you will get child maintenance (too many tales of it being shit/non existent), but see it as a bonus if you do get it.
Find a good nursery/childminder because, in a strange way, they will become the other parent.
The stigma, I've found, is subtle, but still remains in small ways; the raised eyebrow, the fleeting glance, but most parents (esp mothers) look at you like you're superwoman and say they don't get how you do it. I usually reply to that with "I don't have a choice, so we crack on".
My child is only 1, so haven't had the "why don't I have a daddy" chat yet, so will leave that advice to others (and read with interest).
My one piece of advice, especially if you don't have family nearby, is find your tribe. The friends you can call on at 2am and they will come and 'rescue' you, the ones you can meet for coffee and talk about adult things with, the ones who will always check in, even if it's just a silly meme that makes you laugh. You'll need them.
Also, get organised. Planners, wall calendars, alerts/alarms on your phone. Whatever it is. When you're doing three people's jobs (mother, father, paid work), something will slip occasionally. Don't beat yourself up about it. Tomorrow will still dawn and you start again.