Common vs Civil Law

[Conradus]

To sum up, yes, international law trumps national law, speaking strictly legally, but international law is unenforceable. There isn't a legal body you have to answer to, unlike in national law.

It depends on the viewpoint. Legally, there are (here in Europe that I know of) two theories. The first states that international law trumps national law, if that international law is clear enough of itself and doesn't require any further national implementations. Also, it had to be the wish of the parties to the treaty that it should trump national law directly. Belgium's highest Court subscribed to this vision in 1971.
A second theory states that international law only comes into effect when it is ratified in a national law. Later national laws could change that law, and, since it is just a normal law, it cannot trump the higher national sources of law (such as the constitution). Here there is no precedence for international law. The Belgian Constitutional Court follows this theory.

That said, most international bodies don't issue binding laws/regulations. The EU and Council of Europe (with the ECHR) are notable exceptions. As is the UN Security Council in certain, very limited matters. There are very few others.