Does this sound like a bursted/frozen pipe or what do you think is going on?

Likely there is a valve in the basement that will isolate that particular hose-bib and you can just close it now. When the pipes and ice thaw the leak will not continue. If it does, you haven't isolated it. You will then have to shut the whole house water supply off until you can get a plumber to figure it out for you.

Most homes in cold climate ares with exterior hose spigots/bibs have an inside shutoff valve for that spigot/bib to prevent it freezing up and cracking open in cold winter temps. Check the water line going to the spigot/bib (and any others) and see if there is a shutoff valve near where the pipe exits the house to the bib. If so, then close the valve and any others....open the outside spigot and make sure no water still trickles out. In some cases., that valve may not fully close/seat which still allows a slight amount of water to stay/flow into the hose bib and can still crack/leak in very cold temps. You may need to replace the shutoff valve or internal seal to prevent that or at least, cover the exterior bib with insulating boxes to lessen the chance of the bib freezing...insulating boxes sold at home centers that just slip over the bib and held in place with a rubber band or adjustable strip.But they may still not prevent freezing in very cold temps.

https://www.google.com/search?q=hose+bib+insulator...

Another option is to replace the hose bib(s) with a newer design called a frost proof hose bib that is a short section of pipe with a spigot on the end and the shutoff valve part of the spigot is located inside the wall of the house while the spigot part of it still stays outside. That keeps the water from freezing in the outside part of it. The bad part is redoing the piping to attach it to which can be a pain with steel or copper piping. But with that style hose bib/pipe, you don't have to turn off the water valves inside the house every year. In the long run, they are are better option for exterior bibs and more common/standard on newer construction.

https://www.google.com/search?q=freeze+proof+hose+...

sounds like the outside water tap/faucet was not shut off from inside the building (using the knob/lever on the inside of the building) and the water inside the tap/pipe froze, it expanded and bust the tap/faucet and maybe even the pipe, the water dripped out and froze itself shut & that's why its not still dripping outside (hopefully its not still dripping inside your walls) first thing you should do is find where that water pipe comes into the house (usually the pipes are in the basement & this is easy to do if you have an unfinished basement or crawl space) go into the basement & find were the pipe goes out though the wall and follow that pipe back and it should have a shutoff valve/knob/level. turn it to the off/closed position.(sometimes they are not labeled for closed or off. if that's the case then: if its a knob turn clockwise and thread it in as far as it will go, if its a lever turning it perpendicular to the pipe is usually OFF) then you will need to replace the busted tap/faucet and possibly part of the pipe itself. this can wait till spring as long as you have the water shut off to that tap.

if you don't shut the water off to that tap/water line, when it thaws you will possibly have a huge mess with water everywhere, including inside your walls.

if that waterline doesn't have a shutoff to that specific faucet you will have to keep following the waterline back until you find the shutoff valve/know/lever & turn it off there. this may also shut off the water to a portion of the house that that water line is for. each outside faucet is supposed to have its own shutoff valve within 10 feet or so of the outside faucet back that's not always the case in some homes (especially older homes) and as a result you turn off water to a couple rooms by doing so.

once the waterline is turned off you can replace the faucet or best to call a plumber and have them replace it property. it that faucet doesn't have its own inside turn of valve, a plumber can install one for you when he fixes the faucet.

There should be a shut-off valve, either a lever or a turn knob, near where the tap pipe goes to the outside. Shutting that off should eliminate any more leakage. However, leaving it as is right now may be wise because most pipes burst because of trapped air, not from the water freezing, so it is always good idea to keep water (which is comparatively 'warm') moving through your pipes by keeping all your indoor taps dripping through the worst of the temperatures....a few dollars in water usage is way cheaper than having to replace pipes in the middle of winter. If the dripping water drives you crazy, throw a towel over the tap so the water will wick down the towel.

And keep your heat cranked up at night because whether you have a full basement or a crawlspace, your house's heating system will help keep the area warmer so your pipes have less of a chance to freeze.