This can apply to a broader period, but an answer like "we can say at 18xx, steamships overwhelmed sailships in number" would be a better answer. And I know "hybrid ships" ran for a while, but was the period they needed sails on a steamship long?
The first Royal Navy ships with steam propulsion started to be built in the 1830s - HMS Messenger, purchased in 1830, was the first such ship to enter service (though she was a transport, rather than a warship). The first warship was HMS Dee, built in 1832. Steam propulsion began to take off in the 1840s, with most of the fleet's smaller vessels being steam-powered. The first propeller-driven steam ships were built from 1843, with HMS Rattler decisively showing that it was the better option in an 1845 trial against the paddle sloop Alecto. By the end of the 1850s, the majority of the RN's ships of the line would have been converted to steam propulsion. However, they would retain full sailing rig. This would be retained on all ships of the RN until the launch of HMS Colossus in 1882. Commissioned in 1886, she was also the RN's first steel ship, and the first with electric lighting throughout.
As to when the last steam propelled ship of the RN was built, you could argue that the answer lies within the 20 year rule. Nuclear propulsion uses the heat of the nuclear reactor to create steam, which turns a turbine, in turn causing the propeller to spin. So the last steam ship of the RN would be HMS Artful, launched in May 2014. However, if you're referring to the last ship propelled by coal, then the answer would lie with the dreadnoughts of the First World War. The R-class battleships were designed to be able to use both coal and oil, though their predecessors, the Queen Elizabeth class, burned only oil. Oil firing, which still used steam to propel the ship, would linger until much later. Gas turbines, which are very similar to jet engines, were introduced to the fleet as a sole powerplant with HMS Bristol, and the frigate Exmouth in the early 1970s. They would become commonplace across the fleet in the 1980s, with the Type 42 destroyer, Type 22 frigate and Invincible class carrier.
Since the Royal Navy side had been answered, I'll handle the merchant marine.
Based on Lloyd's steam ship tonnage overtook sailing ship tonnage sometime around 1860s (if Time's "Ships" coffee book is accurate).
However sailing ships remained fairly common until the end of the First World War, as they remained viable as low-cost coastal freight carriers. Cutty Sark for example, the famed clipper, served in this role until 1922.
It was the destruction of so many of the sailing vessels during this war that finally resulted in their general extinction, as the sailing ships lost were replaced by steamships.
Edit: Grammar.