How do you know when garlic bulbs have gone off?
Garlic bulbs can last quite a long time provided they are stored correctly in a cool place with plenty of air circulating around them. If your garlic is stored in the fridge the humidity of the fridge enviroment may cause it to go brown or slightly mushy, in which case it should be discarded. Sometimes also the garlic will sprout green shoots. If the garlic is sprouting the shoots will taste bitter and should be removed if you want to use the garlic cloves. Some people prefer to throw the garlic away at this stage as the cloves can sometimes taste a bit bitter too.
Nigella often uses garlic flavoured olive oil, bought from a supermarket, as a convenient time-saver. Don't try to make your own garlic oil as there is a risk of the botulism bacteria growing.
If you don't use garlic that often then you can make a puree with a peeled head of garlic and a very small amount of water and freeze this in portions (a rounded teaspoonful will be roughly one clove). When you want to use the garlic you can just take it out and add it to onions or another dish when you start to cook it. Be careful about putting the frozen garlic direct into very hot oil though, as the oil may spit as the garlic thaws. The garlic may loose a little pungency if it is frozen, but the convenience can make up for this.