In the world of Bushcraft most folk go for the Gransfors Bruks small forest axe, it is recommended by Ray Mears which no doubt helps it's popularity, it is like Rays bushcraft knife, a bit of a "jack of all trades, master of none" compromise and I personally think there are better options.
Axes always tended to be designed for use two handed and have long (around 30-33") handles or one handed use and have short (around 14-16") handles. One handed axes are often called hatchets whilst the longer handled axes of various designs were for felling, limbing and splitting. The Small forest axe is sort of between these at 19" so you can just about use it two handed though it never feels right or you can use it one handed but the long handle stops it pivoting nicely when held close to the head and used for the sort of controlled carving that I do a lot of.
If I were to only have one axe (a terrible thought) it would be a Gransfors Bruks Swedish carving axe. This axe was designed by Wille Sundqvist the Swedish mastercraftsman who first inspired me in my work and it is specifically designed for one handed carving. It will do everything though. It will fell a tree at a pinch and split it into firewood but if you do a lot of that I would strongly recommend an axe with a 30-33" handle. This one excels at carving objects with curved surfaces like spoons but is also very good at hewing flat surfaces to make beams, given time and a woodland it would be possible to build a house and it's contents with this axe.
Managing Director at Gransfors and it looks like we may get the grind returned to the original, watch this space)
Another axe from the Gransfors stable that I rate highly is the wildlife hatchet, these are quite good value at around £50 the handle is 14" same as the carving axe but the head is around half the weight at about 1lb. Anyone can use this axe and it is the only axe I would consider carrying in a backpack. Again it would be possible to fell and limb a tree with this axe though it is perfect for light carving work, such as spoons or tent pegs. When choosing an axe I suggest you use the heaviest axe you can carve with for 20 minutes or so without getting at all tired. If you can manage a heavier axe the weight will do the work and you don't have to swing as much but a lighter axe moves more quickly and lots of small cuts remove wood just as a few large ones do.
Bahco axe looks about perfect, an 800gm head on a 14" handle.
Update 17/5/11 I bought a Bahco axe and it was OK but also needed significant work with a file to get the bevels set and a good cutting edge, it is a good head weight and shape for carving but needs some work.
Another axe which looks good value, I have heard good reviews of but have not picked one up yet is the husqvarna hatchet These retail just over £20 but come with sheath and sharp. I have not had one yet to know whether the bevels are good for carving as they come but it sounds a good option.
Update 10/2/13 I have the Husqvarna now and it is a useful general purpose axe but not great for carving. It came with a rough convex edge so not as blunt as the Bahco but still would need file work to shape the bevels properly before sharpening. It looks quite nice and the handle is much nicer than the Bahco but it is pretty heavy at 780g compared to 660g for the Gransfors carving axe which many find too heavy.
Another nice option is to look round your local car boot fair and buy an old axe. All our grandparents generation had hand hatchets for splitting kindling, at boot fairs they tend to have loose handles and be completely blunt so need a new handle and a regrind but it is a joyful job to bring one of these old axes back to life. Here are a few typical ones. And blog posts on making and fitting a new handle here
This is the sort of carving I tend to do with axes, the axe does matter but correct technique is much more important. Video currently without sound but still shows the techniques.