Trinitron (aka aperture grille) tubes are no more likely to have problems than "regular" (aka shadow-mask) tubes. In many respects they are better - usually geometry. Sometimes colors are not as vibrant.
Geometry problems include pin cushion/barrel distortion (vertical lines at edges of screen bow in or out), trapezoid/keystone distortion (top of screen is narrower/wider than bottom), aspect ratio distortion (squares look like rectangles or circles look like ovals).
Other problems to look for include mis-convergence (red, green, and blue not properly focused together, giving a colored "halo" to text or edges of lines), general focus problems (overall fuzziness), poor voltage regulation (display gets bigger/smaller depending on image being displayed), moire' patterns (looks like a shimmering or wavy pattern at certain resolutions/color depths, usually with medium-dark colors)), and uniformity (even brightness from center of screen to edges).
There is specific software that can generate test patterns but you can check this stuff easily by yourself. Window borders make good horizontal/vertical lines to check most geometric distortions. Set your desktop background to a darker color, then open a good-sized Explorer window with a white background and see if the display gets bigger to check voltage regulation. The monitor probably has controls to fix most of the geometry problems. Convergence, focus, moire', voltage regulation can't usually be fixed by an end-user.