Dermatomyositis: This is an inflammatory disease characterized by skin rash and muscle weakness. The rash often appears as a heliotrope rash on the face and a Gottron's sign, which is a rash on the joints that resembles a rug burn. The presence of joint pain and a rash that looks like rug burn makes dermatomyositis a strong candidate for the single most likely diagnosis.

Psoriatic Arthritis: This condition combines the swollen, scaly skin of psoriasis with joint pain and arthritis. The rash associated with psoriatic arthritis can sometimes resemble a rug burn, especially on the joints.
Lupus: Systemic lupus erythematosus (SLE) is an autoimmune disease that can cause a variety of symptoms, including joint pain and a characteristic butterfly-shaped rash on the face. However, lupus can also cause rashes on other parts of the body, including the joints.
Rheumatoid Arthritis with Associated Skin Manifestations: While rheumatoid arthritis primarily causes joint pain and swelling, some patients may develop skin manifestations, including rashes that could potentially resemble a rug burn.

Septic Arthritis: This is a serious infection within a joint that requires immediate medical attention. While the primary symptom is usually severe joint pain, a rash could potentially be present if the infection has caused significant inflammation or if there's an associated cellulitis.
Gonococcal Arthritis: This condition is caused by a gonorrhea infection that has spread to the joints. It can cause severe joint pain and a rash, making it a critical diagnosis not to miss due to its potential for serious complications if left untreated.
Lyme Disease: Caused by a tick bite, Lyme disease can lead to joint pain and a characteristic "bull's-eye" rash. Although the rash doesn't typically look like a rug burn, Lyme disease is a diagnosis that should not be missed due to its potential long-term consequences.

Erythema Multiforme: This is an immune-mediated disorder that can cause skin lesions, sometimes resembling a rug burn, and can be associated with joint pain in some cases.
Sarcoidosis: A condition characterized by the growth of granulomas (inflammatory cells) in various parts of the body, including the skin and joints. While rare, it could potentially cause both joint pain and a rash.
Relapsing Polychondritis: An autoimmune disease that causes recurring episodes of cartilage inflammation, which can lead to joint pain and, less commonly, skin manifestations including rashes.