I think there are more options that that, but they all require political will and a lot of money.
Non-payment, or avoidance, of child support to result in a criminal conviction possibly leading to imprisonment, much as they do in other countries. Obviously each case would have to go to court as there would be extenuating circumstances in some cases, but that's no different than is the case for other criminal offences ATM, and TBH I'd consider financially neglecting your child as a worse offence than shop lifting or speeding. This would probably be quite easy to sell to the public, despite the inevitable backlash from people who claim that many women are falsely claiming paternity/witholding access (some, of course, will be, but they are a statistical minority in comparison to the genuine cases where single parents are literally left holding the baby with no financial support from the other parent).
More PSHE type education in schools and a greater focus on what a baby really means. Anyone remember the experiment in the US where they gave an electronic doll to the children? It had a huge effect on reducing the number of girls whose primary ambition was to have a child. Again, despite the backlash from those who will claim this is a parent's job, not a teacher's, I think this would be sold to the public quite easily, depending on the spin put on it.
Far greater intervention from the likes of the police and social services in cases where parental conflict or poor behaviour form either parent impacts negatively on children. Bad parenting creates huge social damage. This will probably be unpopular and popular in equal measure. Some will feel it's a gross invasion of privacy. I think if we want a better generation, we need to raise them better. Given the stats on child abuse, neglect, and abandonment, I think it's worth it.
Most single parents are actually mid-30-somethings out of a long-term married/cohabiting relationship, but popular opinion presides that they are jobless 17-year olds. Playing to that, I'd like to see a greater focus on education generally. Raising the age to 18 for full time education is a step towards this. It won't achieve anything though until we get rid of the idea that only academic qualifications are worth having (FWIW most of my peers who got a trade are significantly out-earning me with my 1st class degree and MA). Let's start pushing youngsters - particularly female ones - into vocations, rather than open-ended academic qualifications that don't mean much unless you are prepared to aggressively pursue a career, probably move away, etc etc - all those things that young women in deprived areas who traditionally choose to have children as a type of 'career' - will shy away from.
I could say more but I'm probably boring everyone now.
I am a very lucky single parent. I am in the 1% who own their own home. I still fell foul of a co-parent who abdicated his responsibilities. I lived with him for 6 years before we had our children. I never thought him capable of treating me so badly, let alone our very much planned and saved-up-for DC. Despite all those best laid plans, I still found myself at the mercy of the state when it all went wrong and he refused to pay (self-employed so on a hiding to nothing with the CSA despite holidays and cars). I had a full-time job but with two children to support, a house to find and full-time childcare to pay for, I could not have done it without state assistance towards my childcare costs. As it was I regularly did not eat for the first two years.
Despite all that, and despite hardly bothering to see them (the last time was over 6 months ago), he still has the legal right to override my choice of school for the DC, my religious preferences for them, to prevent me from taking them to see family abroad for an extended period - he still has full parental responsibilities (interpreted in his eyes as rights).
Until people like my X face the very real threat of prison for abandoning (and yes I choose that word deliberately) their children in the same way as I would have done had I just left them at home on their own and walked out, nothing will change. THat's the biggest thing that needs to change.