I was born in Indonesia, where a surname isn't mandatory so I only have a given name, let's say "Samirah" on my passport. I am now in Belgium, in the last year at college, where I signed up using only a given name. I need to find an internship in Luxembourg, but I am confused about how to list my name in my CV or LinkedIn. I'm considering using my father's name but I don't know if this will not be a problem in the future and I'm worried about using "fake information".
Both LinkedIn and your resume should typically always use the name or names you use to identify yourself with. If you only have a given name and no surname, that's the name that should be on your resume.
To use a fictional example, Milla Jovovich's character from The Fifth Element's full name was "Leeloominaï Lekatariba Lamina-Tchaï Ekbat De Sebat". The name she actually used and what she was called by was "Leeloo". That's usually called your "assumed" name or simply "the name you go by". Official documents like an ID card or Multipass if we're sticking with the movie theme don't have the room for long names and so also use a standard variation, commonly the first given name and the last section of the surname to arrive at a "standard" name, for instance "Leeloo De Sebat". If Leeloo were to apply for a job all her application materials would use that shortened name while her actual hiring materials, contract and (being an alien immigrant) her work permit would use her full official name.
But you have rather the opposite problem. Firstly, a name is important in business and it's what you build your reputation and contacts with, so you indeed don't change this lightly or at a whim. But doing so is not uncommon or frowned upon and plenty of people end up partially or wholly changing their name throughout their professional careers. For more on that see the following questions:
- Changing/appending last-name to avoid name duplication
- How do I handle my last name changing after marriage?
- Risks in future career opportunities because of a legal name change
In your case, you'll need to apply with the name you go by, which is currently just your given name. If you have plans to adopt a specific surname or if you actually already have one that you use but just isn't official or listed on your passport, use that to form your full name. As mentioned in the above questions you'd use the name you go by in any contact with a potential employer until you get to the offer stage. At that point you'd simply say something like: "By the way, while I go by X the name on my ID/passport is actually Y." Since you're going to need a work permit or something similar it is absolutely vital that you are clear with any prospective employers that your official name is just your given name. Your hiring paperwork will need to use the name on your passport and discrepancies can cause annoying delays in that process.
Beyond that, it would indeed be helpful to get your name legally changed to use a surname. It's less important if your first name is fairly unique but even then some people will subconsciously think you're less polished or educated because you only have a given name. A distinctive name can be an asset sometimes but generally you want people to remember you because of the excellent work you've done and the skills you possess, not the fact that you don't have a surname. So if possible I would suggest changing your name legally, but for that you'll need to figure out what your country's policy is on that.