Singapore is hot, humid, and opulent. Luxembourg is cold, windy, and opulent.

By some measures, Luxembourg is the richest country in the world.
It is also one of the safest.
It is the only duchy, that is, a country ruled by a duke (a level below a king), in the world.
It is also one of the smallest countries in Europe.

The best way to see is to get in early via a direct train from Brussels to Luxembourg 5:30 AM -> 9 AM for about 26 Euro.

A day is more than sufficient to see major items in Luxembourg, so you can come in the early morning and leave at night. There is a luggage storage facility at the Luxembourg train station, which is open till 9:30 PM.

I would recommend starting the day with a walking tour to cover the major sights of the city.

The city is charmingly beautiful and has a characteristic gray shade everywhere.

I visited the National Art & History Museum, but I would not recommend it, as it was basic and all descriptions were only in French and German.

I would recommend visiting Lafayette Gardens shopping mall to get a great view of the city from its top floor.

Since the city of Luxembourg is on a hill, I would recommend taking the free escalators. There are two of them. So, take one of them down and the other one (the transparent one) to the top.

I decided to take a train + bus to Vianden Castle, which is about ~2 hours by public transport from the city. It is a beautiful castle that reminded me of Sintra, Portugal. If you are short on time, feel free to skip it. This castle is not original and has been heavily restored.

The ticket costs 11 euros for adults. I would highly recommend taking the 2 Euro audio guide at the castle as well.

Note

1. All public transport (including trams) is free in Luxembourg.
2. There are quite a few free bathrooms near the city center.
3. Luxembourg definitely feels expensive compared to most western European cities.