In a perfect world growers would all be using distilled deionized water at with zero electroconductivity (TDS) at a pH of 7.0. As someone who works with growers to optimize their growers I have run into all different conditions. From high mountain water sources which are near perfect, to high sodium to highly alkaline calcium carbonate sources. There is no solution for everyone and it is ultimately up to the grower on how to adapt their feeding schedule to the local conditions. If there are potential problems the grower does not want to go in blind as there will be problems and yield will be sacrificed.
Most hard water is the result of high levels of calcium carbonate and less commonly magnesium carbonate or calcium sulfate, or some combination of the three. This effects hydroponics in a couple of different ways, firstly the carbonate ion is an alkaline buffer and so can require a lot of pH down to lower the levels to the optimum levels of 5.8 to 6.2. Higher levels of calcium can increase the likelihood of a reaction happening with phosphorus and lockout occurring. Adding a phosphoric acid based pH down will increase the chances of this happening since the more phosphorus is being introduced.
For high levels of calcium a hard water feeding program should be used. In a three part feeding program the Micro component usually provides around 5% nitrogen and 5% calcium using calcium nitrate as the source. A fertilizer designed for hard water will still have the 5% nitrogen but will lower the calcium to 1% to decrease the chance of a lock up. A different source of nitrogen is used to bring the levels back up, usually ammonium nitrate or urea (urea is not recommended in hydroponics because it needs time to break down).
A pH down other than phosphoric acid should be used as well. Nitric acid is the best since it's a strong acid and also provides a nitrogen source, but unfortunately is also very dangerous and so most fertilizer manufacturers don't want to handle it. Sulfuric acid should also be avoided as it can also cause fall out, and is also dangerous. I realize that doesn't leave very many options, unfortunately.
Fortunately this problem is way less common than hard water occurring in coastal areas or places which were once covered by sea. The sodium can either occur from chloride (table salt) or bicarbonate (baking soda). The chloride is more toxic but the bicarbonate causes instability in pH. Having water with high sodium is really bad and can literally kill your plants. Consider a different water source because growing with a high sodium level is a fight that you can not win.
Water that has high levels of sulfur is characterized by a rotten egg smell. The good news is that it isn't dangerous to plants and won't upset a feeding schedule. Since it is generally associated with sulfur bacteria which break it down as an energy source there can be problems caused with filters and emitters plugging on water systems.
The problem isn't too hard to fix, a carbon filter will remove sulfur compounds. If this isn't an option keeping it in a holding tank with aeration can also drive out the sulfur. If the water comes from a well which has high levels of sulfur then shocking it with chlorine should take care of the problem.
Like sulfur, iron isn't really harmful it's more unsightly causing brown stains on things that the water comes in contact with. It can also cause filters and emitters to plug either from particulate or from bacteria using it as an energy source creating a biofilm.
Unlike other elements on this list it isn't detected by "eyeballing" as it's colourless and odorless and doesn't form scales or fall out. It can really only be tested by sending a sample to the lab. While low levels are not going to have a huge impact on yields arsenic will accumulate in plants and get passed on to any people or animals who consume them.
Again, not necessarily detrimental to crops but it is a sign that sewage is contaminating the water supply. Consuming plants contaminated with coliforms increases the risk of disease in people and animals.
What to do
Know your water source
If you are from a major city the chances are pretty good that your municipal government publishes water quality data online and a simple glance will tell you what you're dealing with. In addition there will be treatment to prevent any contamination from bacteria or heavy metals.
For those who are outside of a major centre more diligence will be required. Sending a water sample to an environmental lab will breakdown exactly what you are dealing with. When reading the test results the readings are typically in mg/ml or ug/g, these are these same measurement and coincidentally also works out to parts per million.
Unsatisfactory water is put under high pressure through screens which don't let the contaminants pass through, only clean water. The screens are quite delicate and expensive so extra steps should be taken to minimize the contaminants going into them. First off a water softener will trap most of the calcium in the water and replace it with sodium which is easier for the screens to remove. From there it should go through a 10 micron filter to remove any particulate and finally a carbon filter which removes chlorine as it is very destructive to the screens.
Water quality when cycling through a reverse osmosis system is going to vary depending on the quality of the water going in and how good of a system is being used. It should make a huge difference to all of the aforementioned issues. Keep in mind that it is wasteful, typically wasting one liter for every liter produced. Much more waste will occur if the unit is not running properly as it will prioritize waste before it will sacrifice quality.