Very few ships were lost on D-Day itself. Not counting the countless landing craft and similar small craft, only two ships were lost on the 6th June 1944. The British-built Norwegian destroyer HNoMS Svenner was sunk by torpedoes from a group of German torpedo boats, while the American destroyer USS Corry was sunk by a mine off Utah Beach. In addition, the British destroyer HMS Wrestler was heavily damaged by a mine, and while recovered, would not be repaired.

If we extend our search to operations off the Normandy beachhead during June 1944, then quite a few more ships were lost, or heavily damaged. However, due to the sources I have available, this will be restricted to Royal Navy losses. Mines were probably the biggest threat. The battleships Warspite and Nelson both took heavy damage from mines, on the 13th and 18th respectively. Similarly, the cruiser Scylla was heavily damaged by a mine explosion on the 23rd, and was not repaired. Two destroyers were sunk, or rendered beyond easy repair by mines - Fury on the 21st and Swift on the 24th. Three more would receive similar damage from mines off the beaches in July and August, as would the minesweeper Chamois. The corvette Pink would also be lost to a mine on the 27th.

The threat of air attack was also present. The destroyer Boadicea was torpedoed by aircraft on the 13th June. The frigate Lawford, which had been converted to a headquarters ship, was hit by a German glide-bomb and sunk on the 8th. However, Allied air attack was actually a bigger threat to British ships - three minesweepers, the Salamander, Britomart and Hussar, would be sunk or damaged irreparably off the Normandy beaches by an attack by British Typhoon aircraft on the 27th August. Submarines were an equally pressing threat, with HMS Quorn being sunk on the 3rd August, possibly by a human torpedo. The frigates Mourne and Blackwood would both be sunk by a submarine on the 15th June, while HMS Goodson was irreparably damaged by a torpedo while operating off Cherbourg on the 25th. The minesweepers Magic, Cato and Pylades would be sunk by torpedoes, possibly human, in early July. Two Canadian corvettes, Regina and Albernie, would be sunk by submarines in August. Surface attack was considerably rarer than any other threat. Beyond Svenner, only two RN ships were lost to German surface threats in the Channel in the period of interest, both frigates. Halstead was lost in an action with German E-Boats on the 11th June, while Trollope would be lost to E-Boats on the 6th July.

Source:

H. M. SHIPS DAMAGED OR SUNK BY ENEMY ACTION 3rd. SEPT. 1939 to 2nd. SEPT. 1945, C.B. 4273(52), Admiralty, 1952