If the particles are rubbery, oily, and smudgy in texture then they are most likely from the disintegration of rubber materials used in plumbing fixtures. The first places to look would be at your plumbing gaskets, hot water flex tube and o-rings as they can disintegrate over time and can collect in toilet tanks, tubs and around faucets. Depending on the case you will want to drain your hot water heater to remove any remnants of the rubber particles.
The city currently utilizes chloramines as a disinfectant and it is known to cause disintegration of some types of plumbing fixtures overtime. If you experience frequent disintegration of your o-rings and gaskets contact the manufacturer, or replace with plumbing fixtures that are resistant to chloramine. Be sure to take off the tubing after taking all precautionary/manufacturer suggested steps as the outside of the hose or tubing may look in tact and the inside may be disintegrating.
If the particles are small black flecks they are most likely from filters that you have hooked up to your house that uses carbon filters to remove contaminants. If you do have filters and determine that this is the cause, you should change or flush your filters per your suggested manufacturer guidelines.
If the black particles smudge when you apply pressure they are normally due to the anode in your hot water heater needing to be cleaned or replaced following manufacturer recommendations. The hot water heater will then need to be drained per manufacturer suggestion, this should be done on a yearly basis to prevent build up inside your tank.
After you have found and corrected the issue, remove and clean your aerators from your faucets, then flush the system by running cold water for 5 minutes from each of your taps it will likely resolve the issue of black particles. If the problem continues after flushing and you have determined that the source is not a rubber gasket/ tube, your hot water heater or a filter, please contact the Water Quality Assurance Laboratory at 321-433-8709.