It depends on were you live and the type of person that you are. It also depends on the types of questions that you are asking. Many people choose a shul based on a rabbi that they are able to ask. Others will ask their rebbe to suggest who to get in touch with. I know someone whose chavrusa became a rav and who uses him. Some people will ask the rabbi of their shul who he relies on for difficult questions and will use the shul rav for the "simple" questions (such as kashrus). Some people approach the rabbi of their shul because they know that he will send the question on "up the chain" if it is out of his range of expertise.
Some rabbonim are reluctant to get involved with certain types of questions and will reccomend someone to whom to have the questions referred. Some will not answer themselves but will expand the question and call it in themselves.
You should speak to your local rabbi and ask him what he thinks. He will not be insulted, but will answer you seriously and to the best of his ability. If you are in an isolated area and there is no rabbi, then you can call the nearest community for help.
Sometimes calling a central location such as the OU or Yeshiva University or the RCA or Chabad can get you the names of people in your area. For example, a rabbi who has retired may be willing and able to answer questions while not having a congregation. Sometimes a person who has semicha from a Yeshiva who does not "go into rabbanus" will have his name available for consultation.