Urination is no reason to get rid of a perfectly healthy animal. Exhaust all alternatives before you even consider dropping your cat off at a shelter, or worse, putting a healthy and lovable creature to sleep. Remember: This is your family member. We all go through stressful periods in our lives and more than likely that is simply the case with your pet. It may subside on its own, but more than likely this is a manifestation of stress that requires you to pay more attention to what your cat is telling you.
Cat urine is one of the strongest and most difficult odors to remove, which makes it all the more frustrating when your feline consistently urinates on your furniture. There are products you can buy to curb you cat from frequently urinating in a favorite spot, but the most effective strategy requires behavior modification: getting to the root of the problem, consulting a professional if need be, and attacking a very frustrating problem at its source.
Place plastic mats with harmless plastic spikes in a favorite urinating spot. These are the plastic chair mats which have prickly plastic "spikes" attached to the bottom of the mat. Turn the mat upside down so these spikes are exposed. The cat will leave the area alone as long as the mat is in its spot. The longer it is there, the less likelihood the cat will go back to that area. Plus, over time, the cat will begin to associate the area with those uncomfortable spikes, thus making it an unattractive location (and making their litter box far more attractive). This is an effective and safe deterrent.
Spray a synthetic version of naturally occurring pheromones in the target area. Pheromones are actually chemical marking agents, which are used by cats to mark their territory. These sprays have been known to reduce or eliminate this cat marking behavior. You'll want to obtain a product from your veterinary office or animal clinic, so you know it is a quality and legitimate product. It comes in both spray pump and plug-in room diffuser form.
Try different textures of litter. Your problem could be your cat simply doesn't like the litter you're using. Many cats actually don't like the common clay litter. Try crystals or newspaper-type litter. Think about it this way: If you were a cat, what would you like better? Tough, uncomfortable clay in between your toes, or nice plush fabric, carpeting or bedding?
Get to the source of your problem. This is the key. Visit your vet to rule out any health problems. It could very well be Urinary Tract Infection (UTI). If this is the case, it's actually a very easy fix. It can be resolved with antibiotics; however, if the UTI has been going on for a lengthy period of time, the cat may have developed a hard-to-break habit of urinating on your furniture, thus making it more difficult to break the habit even once the physical problem has been eliminated.
Modify the environment. The most difficult-to-crack source of the problem is stress. Your cat may be stressed out due to the introduction of a new feline to the household, a major move or a change in your routine. Try to think about when the problem started, and hone down on what may be the source. You'll want to eliminate the source of stress. If it's your new feline, that by no means getting rid of him or her, that simply means finding new ways to get your old cat to play or socialize with the new one, or separating them for some time to give your first pet the space and control it desires. If you haven't been spending as much time at home, you'll want to make a conscientious effort to give your kitty the attention it needs and is accustomed to.