A bradawl can be stored with all of your other tools, whether that be in a toolbox, shed, or other container.
Bradawls do not require any particular cleaning but may need sharpening if the tip becomes really blunt.
When sharpening a bradawl you must take into consideration the shape of the point, ensuring it is sharpened evenly to maintain this. It is important to ensure the points on all three tips are not too sharp however, as this can cause the bradawl to penetrate the wood fibres incorrectly.
A few different tools can be used to sharpen an awl, including a hand file or whetstone. You can use a combination of increasingly finer sharpening tools to achieve a smooth finish but it is unnecessary to purchase new tools for this – just use what is available to you.
There are several different methods for sharpening a bradawl and it is up to the user to decide which they prefer. Some of the variables include whether to move the bradawl against the sharpener or vice versa and whether or not to use clamps to hold the stationary element.
As long as you go with the shape of the bradawl tip, you should achieve an acceptable result, but if not, bradawls are an inexpensive tool to replace.