This can be a very real issue for the child, who may grow up not knowing who his or her father is. Studies show that a child is likely to have better outcomes in life if they know who their parents are. Growing up not knowing who your father is will have a significant impact on a person's life. Just watch "Missing Pieces" or "Lost & Found" on TV if you need proof!
A more immediate issue can arise when a father is not named on the birth certificate, and will not acknowledge paternity in writing. In that case a mother receiving a sole parent benefit will have her benefit reduced until she is able to conclusively name the father. This is so that the father can then be required to support his child by paying child support. Typically the sole parent benefit is reduced by over $20 a week.
Back in the day all sorts of lurid information was required to prove a paternity claim. These days almost all applications are resolved by DNA testing. The Court can recommend DNA testing is done, although it cannot require it, and no person can be compelled to undergo a DNA test if they do not want to.
If a putative (possible) father refuses to take a DNA test, the Court can draw an inference from that. That is, the Court can assume that the man is refusing to take the test because he knows very well that he is the father, but he doesn't want it to be proven. In that case the mother will need to provide corroborating evidence as well, but it is quite possible for the Court to make a paternity order even without DNA testing.
The application is a relatively straightforward one for an experienced family lawyer to prepare, but there are a number of pitfalls for a person trying to make the application on their own. It is recommended that you consult a specialist family lawyer if you need to make, or respond to, an application for a paternity order.