A common issue for fathers, and especially unmarried fathers, is the lack of a presumption of fair, equal custody, of their children.

What some places are trying to do is define joint equal custody as being in the best interests of the child. There is a Wikipedia article tracking some of this progress here. So far, only the state of Kentucky has passed any kind of equal parenting law.

Another issue is whether or not a father can legally establish paternity without the consent of the mother. Having default equal custody is great and all, but only the legal father can apply for it. If you're married, you pretty much always get a presumption of paternity. But if you're not married, your rights as a father are dependent on whether or not the mother puts you down as the father. AFAIK, this is a problem almost everywhere (see here for example). Kentucky is working to fix this though. Alberta also has a one year grace period where the father can attempt to establish paternity.

A few states have made progress in other areas.

Arizona

In 2013, the state of Arizona passed a law that helps fathers in quite a few ways. Courts are supposed to maximize the amount of time that both parents get with the kids. And parents who try to do things like delay court hearings or "mislead judges" (which I assume means making things up to gain custody) get penalized. This stands in stark contrast to the situation before 2013 where the gender of the parent was directly relevant in custody decisions (with the mother getting preferential treatment).

A 2017 bill would have established that equal parenting was in the best interests of the child, but it didn't pass.

Kentucky

In 2017, Kentucky passed a law making equal custody the default for temporary custody arrangements. This can only be overturned if the parents agree to something different, or if it could be proven that it wasn't in the best interests of the child. This was considered important because initial custody arrangements often influence permanent arrangements.

In 2018, a similar bill was passed for permanent custody, making Kentucky the first state ever to have these types of laws.

In 2019, a bill was proposed that would allow an unmarried father to be considered a parent, exactly the same as the mother, as soon as he can prove paternity. As it is now, an unmarried mother can refuse to let the father establish paternity, which is basically a loophole around the above two bills. Information about that can be found here.

Areas with failed legislation

In Florida an equal custody bill was passed, but then vetoed by the governor due to pressures from NOW and other feminist lobbying groups. There was a very large public backlash against this when it happened (see here).

Something similar also happened in Minnesota.

Most of the Wikipedia article about this is basically a graveyard of failed proposals by various US states. Outside the United States, Canada also had an equal custody proposal in 2014 that failed.

Pending legislation

There are a few US states which have bills currently being looked at. Italy also has an equal custody bill being looked at. And while not perfect, Finland has a proposed bill that would force judges to "consider" 50 / 50 custody.

If anyone has any more information, please let me know. You'd be surprised how difficult it is to track anything like this down. I tried doing something similar for rape laws (whether or not a man can prosecute a woman who rapes him) and couldn't find anything definitive outside the examples given in the reference book of men's issues. Many people, including lawyers, get very antagonistic towards people for even asking these kinds of questions.