I am sorry to hear that your fellow is limping on his front leg, preferring not to bear any weight on it.
Bichons are prone to joint problems because their conformation, short legs that are often twisted rather than straight tends to put more stress on their joints. This conformation can lead to stretched or torn ligaments in the carpal (wrist) joint.
Another possible cause of his symptoms is intervertebral disc disease. This is when the spongy discs between the vertebrae in his spine either prolapse or leak and put pressure on his spinal cord. If the affected disc is in the neck this pressure can affect the nerve roots to his front leg. This is quite painful and can lead to painful muscle spasms or if there is enough pressure then even paralysis can result.
At this point whatever his injury may be I recommend keeping him very quiet, no running, jumping or bending over (elevate his food and water bowls to head height) is best to relieve any stress on his neck and front legs.
If you have a crate for him I highly recommend using it. The less he moves around the more comfortable he will be and the faster he will heal. He should go out on a leash to relieve himself. Do not use a collar for him, a harness which more evenly distributes forces if he pulls on his leash is better.
You may need to confine him for several weeks, even as he starts to feel better or he may reinjure himself. Keeping him on the thin side is recommended to decrease stress on his neck and joints, but is no guarantee that he won't have another episode. Once a dog has one bad disc or joint pain the likelihood of another episode is very high.
If you are interested in reading more here is a link to an excellent article about intervertebral disc disease, its causes and therapy: http://www.petwave.com/Dogs/Dog-Health-Center/Bone-Joint-Muscle-Disorders/Intervertebral-Disk-Disease/Symptoms.aspx
Long term for any joint pain I recommend using a combination of a glucosamine/chondroitin product (examples are Dasuquin or Cosequin) and an omega 3 fatty acid (like 3V Caps or Derm Caps). I recommend an omega 3 fatty acid dose based upon the EPA portion (eicosapentanoic acid) of the supplement as if we do that the rest of the supplement will be properly balanced. Give him 20mg of EPA per pound of body weight per day. For example a 15 pound dog could take 300mg of EPA per day.
Omega 3's and glucosamine/chondroitins work synergistically and improve cartilage health and joint fluid quality and quantity as well as reducing inflammation. They can take several weeks to see full improvement but some dogs do very well with them alone. They are available over the counter.
Another option is a product called Duralactin. This is an anti-inflammatory product derived from milk proteins and it also has omega 3 fatty acids incorporated into it which can be very helpful. See this link for further information: http://www.duralactin.com/products_canine.htm
If she isn't improving dramatically with rest over the next 48 hours then he should be examined by his veterinarian.
If the over the counter supplements I mentioned and rest are not enough her veterinarian can prescribe drugs that are more potent at relieving pain and inflammation. Veterinary drugs we can add include a nonsteroidal like Metacam, Deramaxx, Previcox or Rimadyl. If those aren't enough we can add another drug in the opiod family called Tramadol and/or another drug called Gabapentin.
Over the counter drugs that we use aren't recommended in dogs as they have too many side effects.
Aspirin is the only possible one to use but it isn't very good for pain control, and long term it can create gastrointestinal upset, ulcers and interfere with clotting. If your pup is healthy otherwise you can use aspirin but not for more than 2 to 3 days consecutively and it should always be given with a meal. If you choose to use it watch for lack of appetite, vomiting, blood in the stools or dark tarry stools and stop immediately if you see those. Do not use aspirin if your dog has liver or kidney disease or a history of a sensitive stomach or clotting problems.
The dose for aspirin is 5mg to 10mg per pound of body weight orally every 12 hours (about one low dose 81mg aspirin for an 8 to 16 pound dog every 12 hours). Always give with a meal. Do not use for more than 2 or 3 days.
Be aware if you choose to use aspirin and it doesn't help your veterinarian will be limited on what they can prescribe as there must be a 5 to 7 day washout period between different nonsteroidals or nonsteroidals and steroids.
You can also try alternating warm and cold packs on his painful areas for 10 minutes at a time several times a day.
Please let me know if you have any further questions.