Mesothelioma is a cancer and unless a doctor has a reason to suspect Mesothelioma specifically, they will test for cancer in general.

CAT Scan – Computerized X-ray provides a very detailed picture of the size and location of the cancer.
Mediastinoscopy - Mediastinoscopy involves making an incision approximately one inch above the breast bone and literally carving a space through your body. A scope (mediastinoscope) is then put into the created tunnel which provides a view of the affected areas. The scope may provide direct visualization or may be attached to a video monitor.
Needle Biopsy - Needle aspiration biopsy (NAB), also known as fine needle aspiration cytology (FNAC), fine needle aspiration biopsy (FNAB) and fine needle aspiration (FNA), is a diagnostic procedure sometimes used to investigate superficial (just under the skin) lumps or masses. In this technique, a thin, hollow needle is inserted into the mass to extract cells that will be examined under a microscope. Fine needle aspiration biopsies are very safe, minor surgical procedures. Often, a major surgical (excisional or open) biopsy can be avoided by performing a needle aspiration biopsy instead.
PET Scan - is a nuclear medicine medical imaging technique which produces a three-dimensional image or map of whats going on in the body.
Thoracentesis - also known as thoracocentesis or pleural tap is an invasive procedure to remove fluid or air from the pleural space for diagnostic or therapeutic purposes. A cannula, or hollow needle, is carefully introduced into the thorax, generally after administration of local anesthesia.
VATS - Video-assisted thoracoscopic surgery is a surgical operation involving thoracoscopy, usually performed by a thoracic surgeon using general or local/regional anaesthesia with additional sedation as necessary. It’s historically also been referred to as pleuroscopy.
Wedge Biopsy - An incision is made through the skin, and a wedge of tissue or tumor is obtained and sent to a laboratory for analysis. A wedge biopsy is often used when other biopsy methods have been unable to confirm a diagnosis.

See also:

MESOMARK: a potential test for malignant pleural Mesothelioma. http://www.ncbi.nlm.nih.gov/pubmed/17289801