For level tones, it's pretty easy to list them. Most languages with tones have only a simple high-vs-low distinction; a few more have a three-way high/mid/low system, and very few have more than five. As a result, the IPA provides five different diacritics for pitch level: a̋ á ā à ȁ.
For contour tones, there are a lot of possibilities, but the most common are a simple rise (low-to-high) and fall (high-to-low). Combining these with a few different starting points, you can probably shove most languages' tone systems into this framework if you apply enough force. Something like the Mandarin third tone could be analyzed as low rising, with the initial dip swept under the rug. As a result, the IPA includes only two diacritics for contours: ǎ â.
For register tones aka register complexes…good luck.
In many languages, such as Vietnamese, "tone" includes a lot of other features apart from pitch. The Vietnamese sắc and ngã tones, for example, are distinguished primarily by glottalization; the ngã tone includes a glottal break in the middle. Similarly, the Hmoob (Hmong) "S" and "M" tones are distinguished primarily by airflow, with the "M" tone being breathy as well as low. These features are correlated with tone and best analyzed as part of the same feature, since e.g. Hmoob doesn't allow a high breathy tone, only a low one. To quote mango's answer, "Vietnamese tone pitches and contours cannot be separated from phonation type or manner of articulation of final consonants, so classification based on pitch is not adequate."
In addition, it's not actually that uncommon to see contours that can't quite be analyzed as a simple rise or fall. Contour tones aren't that common among the world's languages, but e.g. there's a reason the Mandarin third tone or the Vietnamese hỏi tone is usually described as having a fall as well as a rise. You could add this "dipping" tone (falling-rising) to the list of contours, and then you'd cover everything in Mandarin, Vietnamese, and Hmoob, but that's just a patch over the problem.
This is part of why the IPA eventually switched from tone diacritics to tone letters, which can spell out any sort of level or contour tone. This allows for a more accurate representation, at the cost of more complexity.
In the end you'll have to decide how much you care about accuracy, and how much you care about simplicity. It's always a tradeoff between the two. The fact that the IPA used to get by with five levels plus two contours is a selling point for that system (adding the dipping contour if you like); the fact that they eventually moved away from it for the more complex tone letters is also a selling point for that one.
