In 1708 Queen Anne refused to grant Royal Assent to the Scottish Militia Bill. This meant that the Bill, despite successfully passing through the House of Commons and the House of Lords, did not enter law.

That was the last time a monarch has refused to grant Royal Assent to a Bill (which is still required in the UK and other countries with the Queen as Head of State - such as Canada, Australia and NZ), and therefore the last time the monarch used a power (rather than just influence) to affect the governing of the kingdom.

The Monarchy of the UK exerted some influence in politics through its Governor-General in Australia in 1975. The impression I get is that this was still a highly contentious move, and the effects of it are not widely agreed upon.

When Gladstone resigned as Prime Minister in March 1894, Queen Victoria did not ask him who she should appoint as his successor but chose her favourite Liberal, Lord Rosebery. (Gladstone would have advised her to choose Lord Spencer if he had been asked.) Whether it was within the Queen's lawful power to appoint a new Prime Minister without taking the advice of her first minister was unclear then but it's certainly clear that the Crown does not have that power today.

There is currently a concern over whether the Prince of Wales (acting in stead of the Queen) is exerting improper influence over the preparation of bills to be presented to Parliament. This is related to a convention under which the monarch must consent to any bill (note - bill, rather than law) affecting the crown. It has been alleged by one MP that he was required to alter the content of a bill.