Many people use tap, filtered, spring, or drinking water for their soap and have no problems. You're going to see many folks chime in here with this kind of answer to your question.
If your water has trace minerals in it, however, especially metals such as copper or iron, you may see spots of DOS (rancidity) forming as the fats in your soap react with these metals and with oxygen in the air. Tap, filtered, spring, and bottled water all contain minerals that either come naturally from the water source or are added after purification to improve the taste -- there's nothing done to this water to remove trace minerals.
Most or all of the minerals in water can be removed by distillation, reverse osmosis (RO) treatment, and deionization. Water that has been treated with these processes tastes "flat" because the mineral levels are so low. This water is valuable for chemical processes where trace minerals may cause undesirable chemical reactions (including DOS).
Rain water is essentially distilled water, although it can contain dust, dissolved acids, and bacteria collected in its travels through the atmosphere. Water collected from dehumidifiers is also essentially distilled, but it too can contain dust, bacteria, etc. unless the collection system is kept scrupulously clean.
I will add that if you make liquid soap (soap made with KOH as the lye) you will probably want to use distilled water to make and dilute the soap. The minerals in tap water will cause the diluted soap to be cloudy. More:
http://www.soapmakingforum.com/showthread.php?t=52456