When a couple who isn't married have a kid, what determines the kid's last name?

In western cultures the kids traditionally got the father's surname, but it's all changing and lots of kids are given their mother's surname instead. It's up to the parents, really.

Usually their mothers name.

There is one aspect about a person born out of wedlock that has always puzzled me.  When they are referred to as  an illegitimate child, they are often referred to in a scornful manner as though their unfortunate existance is some fault of their own.

As they did not even exist until  a short time after some 'wandering minstrel' persuaded Ethel to remove her knickers, how the hell can he/she be responsible in any way for his\her eventual existence ?

 Your opinions would be much appreciated.

Where the couple is in a ‘committed' relationship children are given their surname according to the traditions of the culture they’re living in, but sometimes the parents can choose one or the other or both surnames. (My illegitimately born cousin has his fathers surname, his half siblings have the surname of their mother) 

Where the father is not named on the birth certificate, the mothers surname would be used. 

It is up to the mother, but usually the kid goes by the father's last name. 

With a few exceptions, most states allow parents to choose their child's name, without restriction. Unmarried partners can decide to choose one parent's last name, hyphenate both last names, or create a new last name that combines both parents' names.

In the UK, for a father to be on the birth certificate, they have to attend the registry office with the mother.  However, unless there are big problems in the relationship, it is up to the parents to decide on the surname.  It could be the Mother's, it could be the father's or a double barrelled surname. name.

Even if the mother went to register the birth on her own, she could still choose a different surname to her own if she wanted.

This is from the website linked:

Parents may give their child any forename or surname. No additional documentation is  required  under  UK  law  although  there  is  a  facility  for  doing  so  through  the  UK courts by way of a Deed Poll or Statutory Declaration.The surname given to a child when its birth is registered in the UK is generally, but not always,that of the father. However  it  sometimes  happens,  especially  in  the  case  of  persons  born  outside marriage,  that  the  surnames  of  the  parents,  in  any  order,  are used  (hyphenated or not) to provide a double-barrelled surname.English law places no restrictions as to what surname may be registered for a child in the UK

https://assets.publishing.service.gov.uk/governmen...

The mother chooses in the US. Father has zero legal rights unless married to the mother at time of birth, for which there is a legal presumption of fatherhood (even in cases where not biological father) or unless the mother puts the father's name on the birth certificate voluntarily. If any challenges about who father is, although a father may be able to petition for parental rights after baby born, and get a court order to get DNA sample, since baby inside mother's body before birth, wouldn't be able to get a DNA sample until after already born, so name on birth certificate moot at this point.

That would depend on where in the world it is and if any laws in that country state registration of names........ generally names are a choice , where I live  unmarried parents would both have to attend the birth registration for the fathers name to even be on the certificate and  either surname could be used for the child

It depends on the laws of the jurisdiction where the kid's birth is registered, plus the wishes of the parents.