There is a Golden Axe and it is indestructable. I have managed to get one for each of my Animal Crossing characters in the game.
Once you have it you will never need buy an axe from Tom Nooks shop again.
To get the Golden Axe you have to go through several stages in the game.
It takes quite a few weeks to get because you have to wait for special visitors to come to your town before you can complete each stage:
1. Buy a red turnip from Joan the Boar (she always visits Sunday mornings)
2. Give Wendell the Walrus a red turnip to eat (instead of fish) when he next shows up in your town. Wendell will give you a country guitar or a turban.
3. Give the turban from Wendell to Sahara the camel when she shows up in your town. Sahara will give you a red vase or a massage chair.
(If Wendell gave you the country guitar you can give it to KK Slider at the Roost cafe on Saturday night - he'll give you his pic).
4. Give the massage chair from Sahara to Tortimer the mayor when he is outside the town hall (fishing tourneys/bug offs/special events). Tortimer will give you a special scallop shell.
(If Sahara gave you a red vase you can give it to Redd the fox when he visits your town - he will give you a safe which you can then give to Tom Nook who will give you his pic).
5. Give the special scallop shell to Pascall the red otter when he next shows up in your town. He will ask you a few questions about what you want for it. Go for the answers that include the word "golden" and Pascal will give you the Golden Axe.
This is a link to a site called
Animal Crossing Ahead. The page I have linked includes info about all the special characters. Scroll right down and read about getting the Golden Axe - it is called the "Red Turnip Trade"
The site has lots of useful (true) information on the AC: Wild World/DS game.