Certain microphone situations can be particularly susceptible to feedback. Podium microphones and lavaliere microphones, due to the long distance from the mouth to the microphone, are especially challenging. Having the "talent" (person wearing the microphone) project their voice can help, but some people just can't project well, or it's not sufficient.
In these cases, it's time to try "ringing out" the microphone. This is the process of identifying the problem frequencies for that mic, and cutting them down until the ringing and feedback stops. Here is how I approach this task.
Two things to note: First, you need a parametric equalizer on your mixer's input channels that allow you to adjust the frequency, gain and Q (or frequency width) of each band of EQ (or at least the band that you need to adjust). Most digital mixers have fully parametric EQs that work well for this purpose; inexpensive analog consoles often do not.
Secondly, you need to ring out the microphone for the specific person to be wearing the mic, or in the case of a podium mic, the person that tends to present the most feedback challenges.
Ringing and feedback happens at a specific frequency, and at a very narrow band around that frequency. So we first have to get some idea of what band the ringing is occurring in, so you know what band of EQ you need to adjust for that channel.
To identify the frequency, some digital audio mixers have a spectrum analysis display built in, and you can visually see the frequency that's ringing. Another way is to use a website resource to help train your ear to identify the frequency you're hearing.
Once you've narrowed the frequency range down, pick the EQ band for that channel that best fits the approximate frequency of the feedback. If you're just not sure, make a guess and go with it! You can't hurt anything.
Now, set the Q for that EQ band to its narrowest setting, which is the LARGEST number for that parameter. We want to affect the narrowest range of frequencies possible.
Now, set the gain knob to 0, to neither cut nor boost that frequency band. Have your "talent" start speaking, and tell them not to stop no matter what they hear. (They always want to stop when they hear feedback start.) Turn up the fader until you hear the feedback start, and then back it off until it just stops.
Now, turn up the gain knob on your chosen EQ band to boost by about 6 dB. Start sweeping the frequency knob until you hear the feedback start, and set the frequency at the point where it's the worst. (Feel free to drop the fader down while you're dialing it in to prevent the feedback from getting too bad).
Once you have it dialed in, change the gain knob to cut that frequency, and drop it until it's low enough that you can get the volume you need out of that person without risk of feedback starting.
This may sound complicated, but it really is not difficult to do. A little practice will enable you to ring out a mic very quickly.
It's possible that once this frequency is eliminated, another frequency may start ringing. You can follow the same procedure with a different EQ band for that channel to cut that frequency as well.
Note that the more you cut, the more you'll affect the overall sound quality of that mic. But usually, with a tight Q setting, it's not that noticeable as we're eliminating a very narrow band.