Preschool in Murray Ut. or Surounding Cityz!

Check with the local school districts. I'm not sure if Murray has any preschool programs, but I'm pretty certain Granite District does, and they probably have one in Taylorsville. The preschool programs run by the school districts are often government subsidized, so they can offer high quality developmentally appropriate programs without costing a ton. They also work on a sliding scale. Their programs tend to be 2-3 hours for 4 days a week. Just call the district offices.
And maybe you can work out carpool with another preschool mom near you, too. :)

I am sorry you have had so much trouble finding a preschool. I run a preschool out of my home in South Jordan, about 8-10 miles from Murray. It is a 2 day preschool and I have a morning session from 9-11:30 and afternoon from 12:30-3. I have a few more spots in each one. Check out my blog that may answer a few of your questions and give me a call. If I do not answer I promise I will return your call.
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www.cgardenpreschool.blogspot.com

It may be too far for you to travel, but the U of Utah's Child and Family Development department preschool is excellent, and they do offer afternoons part time. Both of my children went through the program there and we absolutely loved it. it is a true preschool rather than a daycare. The adult/child ratio is great, mainly because the U trains its early childhood ed students there. Mary was the lead teacher in both of my children's classes, and we absolutely love her. They use a play-based curriculum that is active, fun, and effective. Parents are welcome in the classroom at any time, and there are observation rooms from which you can watch and listen without being seen.

All that said, by this time of year, they may also have a waiting list. But we did manage to get back in after moving away and moving back this last year mid-school year.

One of my friends enrolled her daughter in one of the Challenger preschools down closer to where you live and they loved it. Wouldn't have suited my family. A bit too driven and overtly academic and would have pushed my two a bit too hard too fast. But my friend's daughter, who is a very advanced kid and was already reading and doing basic math at age 4, thrived there, and made lasting friendships. Plus, she came out of that program and went directly into first grade at age 5. Not something most kids should do, but in this child's case, definitely appropriate.

Other friends of mine have loved the fine arts preschool at the Virginia Tanner Dance studio at the U of Utah. Again, perhaps too far, but an excellent program, from what I've heard. The reason I did not choose that one for my own kids is that parent observation/participation wasn't particularly welcomed, at least at the time we were looking at preschools, and it was important to me to have direct access to my children's classes. Their policy might have changed since then.

The McKee Spanish Immersion preschool in SLC on 11th E. is also one I've heard good things about.

Does any elementary school in your area have a preschool that isn't already crammed full, or doesn't have too long a waiting list? From what I've observed, some schools get openings within the first few weeks of school.

Hope this helps. Good luck with your search.