As a U.S. conditional resident filing immigration Form I-751, Petition to Remove Conditions on Residence, you must include information that your marriage was made in good faith, and is bona fide (not just a fraud to get a green card). Even if you are not filing with your spouse and are instead applying for a waiver of the joint filing requirement, you should submit as much proof as possible that your marriage was genuine and not made to circumvent U.S. immigration laws.
For more information, see Filling Out USCIS Form I-751, Petition to Remove Conditions on Residence (Line by Line). For more on the waivers available, see What if Your U.S. Spouse Won't Sign the Joint Petition (I-751)?
The following categories are examples of the types of evidence that USCIS would like to see in your I-751 packet to prove that you and your spouse have shared a life together, from the most to the least persuasive.
Copies of birth certificates of your children (with both the immigrant and U.S. spouse's name listed) or adoption records of jointly adopted children are the best evidence that you have a genuine marriage.
However, don't worry if you don't have children! USCIS doesn't expect all couples to start a family, much less in the first two years of their marriage.
USCIS wants to see that the two of you share resources and are living as a family. This includes joint financial accounts as well as the maintenance of a shared home. The more documents you can provide with both your names, the better. (But don't fake it for the sake of the application; see, for example, Should We Open a Joint Bank Account Just to Satisfy USCIS That Our Marriage Is Bona Fide?)
To demonstrate that the two of you operate as a couple, you can provide copies of jointly filed tax returns for the past two years, as well as joint savings and checking account statements, investment account statements, credit card and loan statements, and bills of sale for an automobile or other large purchase.
You should also submit a copy of your lease, mortgage or deed for your family home, and utility bills in both of your names.
If you don't live at the same address, it could be viewed by USCIS as a red flag, but that does not necessarily mean your green card application is destined for denial. You should, however, provide a good explanation as to why you are living apart, as well as plenty of additional evidence regarding the validity of your marriage. Many conditional residents have successfully had their conditions on residence removed while living apart. For example, perhaps one spouse needed to move to another location for work and you intend to follow shortly. Or perhaps you or your spouse is attending college or a vocational training program in another city.
While financial documents are great evidence of a bona fide marriage, you can also provide information that the two of you are members of clubs, religious organizations, teams, and other associations together.
Statements that show that you are covered by your spouse's health insurance or that your spouse is covered by your insurance plan are great proof that you are a couple. The same goes for joint automobile or home insurance policies and life insurance policies that list you or your spouse as the beneficiary in case the other dies.
Additionally, you'll want to provide USCIS with copies of your properly signed estate documents (wills, trusts, and even information on your funeral plans or intended burial location) if you have them. This shows USCIS that you and your spouse intend for the other to inherit your property after death.
You can provide copies of travel documents and itineraries for a honeymoon or a family vacation. If the U.S. spouse has traveled abroad to visit the immigrant's family members, it shows a genuine interest in the immigrant as a person. Also, if you two planned a honeymoon or vacation together, it tends to show that your marriage is not a sham and that you are planning trips together for pleasure.
Don't hesitate to share wedding pictures and (non-intimate) photographs of you and your spouse during your marriage. This demonstrates that you are a happy couple and attend events together! If the U.S. citizen spouse bought anniversary or birthday gifts for the other spouse, show the cards and receipts.
You can submit copies of identification documents showing that the immigrating spouse took the U.S. citizen's name or vice versa. However, it is not necessary that the two of you share a surname to prove that you have a genuine marriage. Plenty of couples decide for a variety of reasons to keep their original surnames after marriage.
While it's not the most convincing form of evidence, you can also submit copies of letters, texts, emails, Facebook or social media posts, and other messages that you and your spouse sent to each other during your relationship, or cards, letters and emails from others that address you as a couple.
You can also submit a personal statement describing how you met your spouse and details of your relationship that prove that it was based on love and affection, as well as affidavits from other people who know the both of you or attended your wedding and can provide evidence and affirm that your marriage and relationship are genuine. See Creating Substitute Documents or Affidavits for Immigration Applications for guidance on drafting an affidavit.
If after reading this and considering your options, you might want to get an attorney's help if you still feel you're short on evidence.