Anytime you do work without any payment until the work is complete, you are effectively extending credit to the party receiving your service. How much credit you are willing to extend will vary greatly, depending on the amount and the trustworthiness of the party. For example, if you are charging $50 for something, you probably won't bother to collect money upfront, whereas if you are charging $5,000 you probably would collect some upfront. But if the party you are working for is a large financially sound company, the number may be even much higher than $5K as you can trust you will be paid. Obviously there are many factors that go into how much credit you are willing to extend to your customer. (This is why credit reports exist for banks to determine how much credit to extend to you.)
As for the specific case you are asking about, which may be classified as a decent amount of work for a small business, I would default to having a written scope of work, a place in the document for both parties to sign, and specify 50% upfront payment and 50% payment at completion. When you receive the signed document and the upfront payment (and possibly even after the check clears), you begin work. I would call this my "default contract" and adjust according to your needs depending on the size of the job and the trustworthiness of the customer.
As for your question about how to deposit the check, that depends on what type of entity you are. If you are a sole proprietor you should ask for the checks to be made out to you. If you are a business then the checks should be made out to your business name. You don't need "in trust" or anything similar because your customer, after paying the upfront fee, must trust that you will do the work you promise to do, just like you have to trust that after completing the work you will receive the final payment. This is the reason the default is 50% before and after. Both parties are risking (roughly) the same amount.
Tip: having done the "default" contract many times in my career, both as a sole proprietor and a business owner, I can assure you there is a big difference between a potential customer agreeing to something in advance, and actually writing a check. The upfront payment definitely helps weed out those that were never going to end up paying you, even if their intentions were good.
Tip 2: be as specific as possible as to what the scope of work will include. If you don't, particularly with software, they'll be adding feature after feature and expecting it to be "included".