Matsue is an underrated city for culture vultures, and one fantastic spot to visit if you're an art lover is the Shimane Art Museum. Located on the banks of Lake Shinji, the museum has a host of works on display from both Japanese and international artists.
Of particular interests are pieces by renowned artists Claude Monet and Paul Gauguin. Monet's "The Rock Needle and the Porte d'Aval" and Gauguin's "Watering Trough" were both painted in 1886, and are just a sampling of the western art here. There are also various examples of Japanese screen art, and iconic prints by Hokusai and Hiroshige.
If you prefer 3D art to 2D art, you'll also find sections dedicated to sculptures, lacquerware, and pottery. Some of the tea bowls and tea containers on display are beautifully ornate and well worth a look.
Along with the permanent collection, the museum often has special exhibits which can be accessed at an additional fee. The list of exhibits and their dates are available on the museum website, so do check it before your intended visit to see if something piques your interest.
Also, if you're an international visitor, bring your passport along when you visit. You'll find that at a number of attractions across Matsue if you bring a foreign passport you get a half price admission discount.