The monarchies in the UK, Netherlands, Denmark and Norway remains popular, and some polls have shown record high support the last four years. This is not going to change unless we get some very very major scandals, which is unlikely.
I don't think we will see a republic in Sweden or Belgium either, and I hope that the Spanish monarchy will survive.
As far as UK is concerned, I actually think it is the safest Monarchy in the world, along with the Japanese.
Republicanism in the UK remains among the lowest in the world, with figures rarely exceeding 20% in support of a British republic, some polls have it as low 13%, and consistent ~70% support for the continuation of the Monarchy. And Some polls have the support for the monarchy as high as 82%, others at around 70 to 76%, another poll has the support for the monarchy from 66 to 70%.
To abolish the British monarchy will be very difficult.
1: Most polls must show a majority for a republic.
2: Majority in the house of commons for a referendum, this is not going to happen.
3: Majority in the referendum for a republic, this is not going to happen.
4: Changing the country's name, changing the pound, remove the royal name from all state institutions. These are just some of the things that must be changed.
5: All of this is going to cost so much money that even many Republicans will start doubting it.
The Queen is the most wonderful, forgiving, non judgmental person I know. Sarah Ferguson speaking in 2011.