Luxembourg is unique to being host to a large number of international and multinational organizations and agencies. Working for these government and quasi-government agencies is prestigious and there are many opportunities for internship. Luxembourg is also unusual in entitling some coveted internship positions to semi-diplomatic status with tax and social benefits and even a salary. This is sometimes called the International Civil Service.
The European Union has various agencies, directorates, organizations and support functions that employ more around 2,000 people in Luxembourg. These jobs are a great stepping stone to a high-paid career.
Several papers have a helpful classified's section.
It is also possible to apply directly with organizations. Research your area of interest and find if there are any organizations with online applications.
If you know anyone in Luxembourg- get in touch! It's not necessarily what you know, but who you know. Friends and family, or any contact you have in your desired business may know of a position. There are also online social networks like Linkedin which allows for people to make business contacts.
To secure an internship aboard, it is usually easiest to find a company that sets-up the internship. Some agencies require a fee to secure a placement. Such agencies include:
AIESEC - is a student-run, non-profit organization which offers paid internships in business and technical fields in around 90 countries and territories, including Luxembourg.
Transitions Abroad - This is an expansive site that includes country-by-country listings of internships.