A few years ago during a cold winter something rather strange happened in our garden and I hope someone in the forums can spread some light on how and why it happened. We keep a bird bath in the centre of the lawn and one morning we came out to find that the water had turned to ice and was raised up in the middle forming a point about 4 inches high resembling a stalagmite! There was nothing above the bird bath that could have dripped and try as we might, we have never been able to suggest a plausible explanation. Does anyone have any clue as to why this point formed?

Was the birdbath of the fountain variety, and was the water accidentally running?

I believe what you are describing could be called "pingo Ice". Here is some ice terminology.

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pingo ice - huge ice lens that forms underneat a pond causing a hill to rise up to 50m high

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Pictures of ice "stalagmites"

The following quote is from here.

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3) We live on an 11 acre horse ranch (Peruvian Pasos) in an area just
slightly north of Grand Junction, Colorado. Since we are technically not
in the city limits, perhaps I should say "Grand Junction" so as not to
offend Mr. Adam. Anyway, I have frequently observed this formation of
"stalagmites" in the dog's water dish and other similar containers. I
have also observed that this occurs under the following conditions:

a) There is an extreme, and very sudden, drop in temperature
with very still air (no breeze);

b) The container is relatively small, allowing for a sudden
cooling and rapid formation of an ice "skin."

c) The water in the container HAD NO ICE CRYSTALS before being
subjected to the sudden drop in temperature, was very still--not stirred
in any way, including by the wind.

Obviously, these conditions can be reproduced in the freezer of a new and
efficient refrigerator/freezer: an ice cube tray containing a small
amount of water; still air; no disturbance of the water; and a sudden
drop in temperature by placing the tray near the back of the freezer.
I might also add that the tray should be placed there RIGHT AFTER THE
SELF-DEFROSTING FREEZER HAS FINISHED ITS DEFROST CYCLE
and will therefore try to rapidly re-cool itself, and will not raise
the temperature again for a number of hours.

More curious is, "How the hell did I wind up here, spending an inordinate
amount of time on an ice cube, when I had an entire evening of chores and
paperwork to do?"

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Wow! I'm impressed! No, Lothruin, the water wasn't running. It was a simple bowl on a pedestal. Thanks a lot for the help!

I see the stalagmite appear in ice cubes all the time, but not nearly as pronounced.

I think stillness of the body of water, as well as relative smallness, contribute to the formation of them.

Water Ice occupies more volume per mass than liquid water. So after the initial "ice skin" forms, the remainder of the water as it freezes has nowhere to go but up. The edges of the "skin" are frozen to the side of the container, so the path of least resistance is the middle of the body of ice & water. Therefore, the middle humps up as the water becomes all solid.