What happened to the adjectival forms of Belize and Singapore in their currencies?

There is no strict rule based on grammar that you can give as there is no fixed rule for how to form an adjective from a country. Who would guess that French is the adjective from France? So if the country and its currency are referred to often people will know the adjective.

If the adjective is easy to guess it will help. For example many countries end in an a and you just add an n such as Latvia > Latvian. This is why we do not use Mosotho. (Note that Argentinean appears to be an exception but it probably comes from The Argentine not from Argentina.)

If the adjective is long (Singaporean) or sounds strange in English (Belizean) it will also be a deterrent.

So overall there are several factors which affect how likely we are to use the adjective.

There is one well known country that does not have an adjective of its own. This is the UK. There is a historical reason for this. Before 1801, Britain and the UK were virtually the same but in 1801 Ireland was added to the UK. As Ireland was only gradually accepted into the union no one ever invented the adjective for the UK. For example they had their own currency until 1826. This means that British was the logically correct adjective for the currency until then.

The reason for going on about the UK is because it shows that there may be historical reasons for using a particular word in a particular country. For example the Russian rouble is not the currency of Russia per se but the currency of the Russian Federation and before that the USSR and the Russian Empire. In short there may be historical reasons specific to a given currency.