The most common undesirable feline behaviors are:
First, let me address training as a technique of behavior modification. In a word—NOT. Only your behaviour is modified.
Don't make the mistake of labeling cat pee in the hall or cat poop on the bed as a "litterbox accident". It is no accident. I've come to believe cats are very literal in their expression of "piss on you." This is their main method of expressing displeasure with your behavior.
Punishment doesn't work with cats. The best approach is to give them a reason to behave properly. This usually means giving them all the attention they want. If they're "pissed off" because you're out of town for awhile, be sure the cat sitter gives them all the attention they want. (My cat sitter visits twice a day for about an hour each time. She pets, grooms, and plays with my cats as well as feeding them and cleaning the litter boxes. And I pay her well because I both value her service and my cats.)
Change in our lives—like moving, a new partner, a new job, or a new baby—can cause us to neglect our cats. Just because they are short and have four legs doesn't mean you can neglect them with impunity. Take the time.
Cats debilitated by illness or age may have problems with incontinence. In this case make it easy for Puss to get in a litter box in time by locating several boxes close to the places where Puss hangs out. Be sure Puss can easily get into the box. Set each box on a larger waterproof tray in case of real accidents.
When cats scratch the furniture, locate a sisal scratching post nearby. In fact, put one in every major room. (Cat "scratching" is actually a form of exercise of the toe muscles. Sisal carpeting is the best material for this exercise.)
Mind your manners, say "please" and "thank you." When Puss does what you ask her, even if it takes awhile, always say "thank you."
Sometimes a cat's misbehavior is aggravated by nutritional deficiencies. Adopt a diet that maximizes nutrition (see my Raw Food with Supplements Diet). You can also use medicinal herbs to support Puss's nervous system.