I have lived in Salt Lake City for a year and a half now and I credit chowhound for introducing me to most of my favorite inexpensive places in town: Red Iguana, Mazza, Thai Garden, Settebello, etc. I even found a mention of Pace's Dairy Ann (I hear the locally-made popsicles are not as special as they used to be, but where else can you buy anything to eat for 32 cents?). However, I am suprised that noone seems to have yet mentioned Nielsen's Frozen Custard.
Real frozen custard is a rare and unique treat for an ice cream lover and summer evenings at either location is a true Utah scene. Sometimes the lines are so long, it can take over 30 minutes to get to the counter. Nielsen's is to other ice cream shops as Settebello is to other Utah pizza places.
The two closest locations (3900 S & Highland in Holladay or 2600 S & Main in Bountiful) certainly do not lack for business and it seems like many natives know about these places, but I still think chowhounders should mention this place for the benefit of people relatively new to Salt Lake. The Bountiful location is nice in a Johnny Rockets kind of way. The Holladay store really could not be uglier but there is an air of authenticity to its horrible layout and shabbiness. There are also stores in Layton & St. George, apparently. The summer lines are worth it. They are even part of the experience. I don't think there is a more delicious, more authentically-Utah place to get ice cream in the state.