Etymologically and technically speaking, concerning is or was used in the past to express being worried about something or someone. You express concerning about someone's health or behaviour/well being etc. And Regarding is more often used in business English to express pertaining to someone or something. Although they all express the same idea, one could linguistically argue that you should use CONCERNING her car if you are worried it is dangerous and will kill her. And to use REGARDING HER car if you want to discuss payments being overdue. If you want to speak about her car being badly maintained and are worried it might kill her one day if she doe not maintain it. Technically you could use BOTH Concerning and Regarding. So it depends on what you are most Concerned about and what you want to give priority to YOUR CONCERN/WORRY. Or just to speak to her Regarding her car.

Regarding: ...her car's problem. It's more natural to say "...her car problem." instead. In that case, "car problem" is used as a compound noun and she posseses her car_problem.

‘…in regards to…’ doesn’t sound natural. I think the preposition ‘…with regards to…’ sounds better. Also, …the car’s problem’ sounds forced. Try, ‘about the car’, or ‘the problem with the/her car.’

All of the options you provided are grammatically correct and convey a similar message. However, some might sound slightly more natural or formal than others. Here's a breakdown of each option: "I'm looking to speak to your sister in regards to her car's problem." This option is grammatically correct but might sound a bit formal and old-fashioned. Many people prefer other options for a more modern tone. "I'm looking to speak to your sister regarding her car's problem." This is a clear and concise option that is commonly used in both formal and informal situations. It's a good choice if you want to convey your message directly. "I'm looking to speak to your sister concerning her car's problem." This option is also correct, but the phrasing might come across as a bit more formal or even a touch archaic. It's not commonly used in everyday conversation. "I'm looking to speak to your sister about her car's problem." This is a straightforward and commonly used option that works well in both formal and casual contexts. It's likely the most natural-sounding choice. In general, option 2 ("I'm looking to speak to your sister regarding her car's problem.") or option 4 ("I'm looking to speak to your sister about her car's problem.") would be the most commonly used and natural-sounding ways to convey your message.

Natural is something like ‘I’d like to talk to your sister about her car.’