Differences between "sledge", "sleigh" and "sled"
Solution 1:
A sledge is normally taken to mean ‘a carriage mounted upon runners instead of wheels, and generally used for travelling over snow or ice’. It shares its etymology and meaning with sled, a form which the OED describes as being ‘chiefly dialect and US’. It does, however, have some meanings that sledge does not have. They include ‘any of various devices made to be towed along the sea bed’ and ‘a kind of river-boat used on the Ohio’. Sleigh has the same etymology, but generally describes a sledge which can be drawn by horses.
Solution 2:
There is a difference among American, British, and others; I'll give (my) AmE interpretation.
sleigh - a vehicle like an open-air carriage, pulled by horses, using runners instead of wheels for use in winter on snow/ice.
sled - a small single person conveyance to slide down hills, usually has runners, but may just have a flat bottom.
sledge - any kind of flat conveyance to be dragged over the ground. These are used for conveyance large loads or injured people when a wheeled vehicle is not available. The references I have seen define 'sledge' almost identically to 'sleigh'. I would never use 'sledge' for Santa's kind of sleigh, nor would I use it for a child's winter toy. 'Sledge' is pretty rare in AmE.
Solution 3:
In the US, here would be images of how I define each sledge, sleigh, and sled, despite all of them having a similar definition in the dictionary.
Sledge:
Sleigh:
Sled:
However, snowmobilers often call their snowmobile a sled, as well:
Note: All images came from a Google image search and are presumably copyright of their respective picture takers.