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Simply because the wired version has been around longer, those will be explored first. Some people prefer using a hand held microphone that's directly connected into the sound system. They feel that it's more reliable and provides a stronger more solid signal than wireless ones. Wired mics are not reliant on batteries, (condensers do need phantom power) so are not restricted by battery life.
One of the main disadvantages of wired mics is that they restrict the speaker's/artist's movement, unless you have an incredibly long cord. And with that long cord comes the probability of it being tripped over, as well as the movement of the performer constantly putting stress on the cables and potentially creating other sound problems. Also with a very long cable there is a loss of signal, mainly in the upper registers. Some people also feel that wired microphones create more feedback noise than their wireless counterpart. Of course that's directly related to the quality of the wireless system.
Wireless mics have been the preference of a large number of speakers, singers and other performers since arriving on the scene. They are considered superior by many for a wide variety of reasons. Namely, you can move around much more easily, so for those who tend to be on the dynamic or more animated presentation/performance side, they are a great boon for that reason alone.
Of course, there's no pesky cable to worry about getting tangled or tripping over. With so many motivational speakers and trainers using a variety of projected visual software to enhance their presentation, this freedom of movement is highly preferable. When it comes to wild rockers/singers, they can obviously move and dance all over the stage with greater ease.
However, bandwidth often has a limited range, so the user needs to be conscious of those parameters or risk losing power. There is a possibility of interference from other radio signals or microphones; this problem has been reduced by the creation of several switch-selectable channels that are synthesized to certain frequencies. Because of potential interference as well as the amount of frequency availability, the number of microphones used in a venue may be limited.
More expensive wireless mics offer increased range, so the potential of dead spots during a speech or performance is reduced. The body pack (power source) for the wireless transmitter can be clipped onto the belt of the performer, or in many instances to the instrument being amplified. For example, it may be attached to a guitar strap or in some cases, directly to the instrument itself, such as a flute or French horn. There also some models that have the transmitter within the microphone structure for those speaking or singing.
Essentially, once all these factors are taken into consideration, it does amount to the user's preference and convenience. There are some people who are still die-hard lovers of the wired mic, but for the past few decades the majority of performers/presenters are in the wireless camp. So the use of stage microphones wired or wireless boils down to the user's choice and available budget. As a general rule good wireless microphones are pricey, whereas mics like the standard SM58 are available for around $100.00.