And would anyone care to discuss any important events that caused the downward trend? Is it purely due to wealth (which seems unlikely to me due to corrupt oil-states), democracy (which again seems unlikely, as many democracies have high corruption) or something else? I recognise these are both tricky questions.
The short answer for Denmark would be since around the age og absolutism in Denmark. It wasn't as low as modern day Denmark but it was decreasing. This was due to a few new laws introduced to make sure the king held absolut power and that his authority was not to be doubted. It effectively removed many of the nobles whom had their own agendas and replaced them with either better nobles, clergymen or loyal and talented commoners.
Later the loss of a lot of territory united the danish people and the country became homogenous with almost only ethnically danes living in the country. This unification through loss meant that danes became more trusting towards each other. Also the country had become a democracy through reforms and had kept the tradition of reforms instead of revolution.
Once again skipping ahead to the 1930s and the great depression effect on Denmark. Instead of squabbling amongst each other forever and letting the people suffer the largest parties came together and made “kanslergade forliget” a compromise starting the welfare state in Denmark. By not abandoning the people and creating the foundation for the modern welfare state in Denmark the idea of solidarity grew in popularity. This became especially true in the last half of the 1900s were the moderne danish state was made. It is stil one of the core values of most danes today.
Overtime this low corruption and unification of the danes have grown into trusting of one another which again made bribes and the likes less needed further boosting the growing trust.
It’s important to remember Denmark haven't really been in many serious wars since the endless warring against Sweden except for 1864 were around 1/3 of the countrys farmland was lost. This relativ peace and stability created decent living condition. Not saying they were the best but enough to make sure rebelions an revolutions like the ones in france didn't happen.
These are the most important things about the danish development of low corruption rates.
Harsh laws against corruption during the Age of absolutisme
Fairly homogenous and unified population
Small country
“Hvad udad tabes skal indad vindes” (what is lost outward shall be won inwards)
Kanslergade forliget (The Kanslergade agrement )
Decent or above living standards
Social reforms
High social mobility
Sources: Mette Frisk Jensen - Korruption og embedsetik