The answer to this question depends largely on your personal preferences, the camera you own, and your budget.
Zoom lenses, especially high quality ones can be remarkably good at shooting portraits or prime lenses are not an obvious choice. You need to look at the details, such as aperture and image quality.
Prime lenses do tend to be much faster, that is have a wider maximum aperture than comparable zoom lenses. This is great for portrait photography as it allows you to produce beautiful out-of-focus backgrounds with a shallow depth-of-field. Wide aperture is also extremely important when you work in low-light conditions.
Image sharpness is another area where prime lenses outperform zoom lenses in the same price range. Primes are usually more simple, contain fewer elements and thus there is less potential for distortions or flares.
The price difference between prime lenses in the same range can be often very steep. As a rule of thumb, expensive models will give your higher quality images, and they will be built to last many years. However, due to metal bodies and thicker glass, they tend to be heavier than the cheaper plastic lenses.
There are three main types here: wide-angle, normal and telephoto lenses.
Wide-angle lenses are great if you want to take a photo of the subject with their surroundings (environmental portraits). They are not so good for close-ups as they tend to produce edge distortions.
Normal lenses, with a focal length of around 50mm, give a similar perspective to those experienced by us humans.
Short telephoto lenses of around 85mm are perhaps most used prime lenses portrait photography. They provide an excellent flexibility of taking close-up photos as well as photographing the whole person.
If you have a APS-C camera body, a 50mm prime lens will also fall into a short telephoto range.
Telephoto lenses allow even more ability to isolate the model from the background. However, they tend to be heavier and much more expensive.
In the list below you will find some of the most popular portrait lenses available today: