A full credit report from the big three will show you prior residences and residency dates (approximate) to confirm they lived in NY. It will also show "public records" which will include tax liens, judgments, and collection items within the US. It frequently shows some cryptic employment info, other names used, birthdates, and info on the social security number, such as status and where it was issued. All of this info can be used to corroborate the information the applicant has given you. Any "omitted" information MAY be cause for concern, or at least warrant additional questions.
Regarding employment verification, unless you look up the company in the phone book, and call the published number to locate a Human Resources person or a Manager (without asking for the person named as supervisor on the application), you run the risk of getting purely false information. It they have no public phone listing, you need to check public records to see if the business is registered in your locale, or can otherwise be confirmed as legitimate, and then call an officer or contact person of record.
Essentially the same goes for landlord references...if they rent from a company, get the reference "through the front door", not just calling whatever number the applicant provides. If the landlord is a private individual, at least check the county records for the actual ownership and try to confirm the tenant information. Depending on your market, you might be amazed at how many people "rent" from friends and relatives.
As to "what matters", IMHO, the credit report is great, and my primary tool. It does take some study however. The actual FICO score is not terribly relevant to me--what I look for is patterns of irresponsibility. Life Happens to people. You don't have a choice with medical bills. You don't have a choice, sometimes, when a bankruptcy takes place. You need to look at what I consider "consumer" issues.
Some examples I see all the time: Two or three cell phone companies with collection items; Victoria's Sec***, Pi**a Hut, K Ma**, even the local library showing up as collection items. ANY of these types of collection items, I won't rent to you. Medical bills, I ignore. Foreclosure/bankruptcy I MIGHT ignore, if A) it was at least two years ago; B) there is not a rash of abusive consumer items that precipitated the foreclosure; C) There are no NEW late accounts or collections since the discharge (and ideally, there is some new positive account information); and D) everything else, especially landlord reference and steady employment checks out "stellar". One red flag to consider is if several credit cards accounts show "reported stolen". This MAY be an indicator of someone just having trouble paying all their bills. You just have to study the whole picture.
Hopefully, this will at least give you a fighting chance of obtaining a tenant that will actually pay. Most of the time.
Good Luck!
PS- don't forget to give a proper WRITTEN denial letter to those you turn down, per the FCRA; and you also need to securely archive the application documents for three years to protect yourself on several fronts, including Fair Housing.