What are the most common bursitis problems?  

Several. Bursitis of the trochanteric bursa (side of the hip), and subacromial bursitis (shoulder) are among the mowst common rheumatology problems seen. They can arise from arthritis, or simply exist by themselves. A cortisone injection, if bursitis exists, almost always gives quick and good relief!
Knees and elbows. People who kneel a lot or lean or their elbows a lot are more vulnerable to getting bursitis. The bursa are external to the joint so bursitis can usually be treated by draining the bursa, treating with anti-inflammatories, and using cushions or stopping the triggering behavior. Occasionally one can get an infectious bursitis which requires drainage and antibiotics.