While all residents in Luxembourg are required to , citizens of certain countries will also need to check-in with their nearest embassy. Essentially it comes down to domestic policy.
Australia and the UK, for example, do not run resident registration systems. Canada, Ireland and the US do not either, but their embassies do have optional registrations, mainly to receive safety alerts. France has voluntary registration for its overseas nationals, which can help with some administrative matters.
Citizens of countries with compulsory residency registration systems--such as Belgium, Denmark, Finland, Germany, Iceland, Italy, Japan, the Netherlands, Norway, Sweden and Switzerland--will need to sign up at their country’s embassy in Luxembourg or Brussels.