Please see this link which describes how to make a death notification and how to apply for survivor benefits which you may be eligible for :

http://www.ssa.gov/pubs/deathbenefits.htm

Social Security survivors benefits can be paid to:

• A widow or widower -- full benefits at full retirement age, or reduced benefits as early as age 60
• A disabled widow or widower -- as early as age 50
• A widow or widower at any age if he or she takes care of the deceased's child who is under age 16 or disabled, and receiving Social Security benefits
• Unmarried children under 18, or up to age 19 if they are attending high school full time. Under certain circumstances, benefits can be paid to stepchildren, grandchildren, or adopted children.
• Children at any age who were disabled before age 22 and remain disabled.
• Dependent parents age 62 or older

See here for details :
http://www.ssa.gov/ww&os2.htm

You can report the death to a service representative by calling SSA toll free, 1-800-772-1213 (for the deaf or hard of hearing, call SSA's TTY number, 1-800-325-0778), from 7 a.m. to 7 p.m., Monday through Friday. Whenever you call, have the deceased person's Social Security number handy. To report the death of a beneficiary, you should:

• Notify the bank or other financial institution of the beneficiary's death.
• Request that the bank or other financial institution return any funds received for the month of death and later to Social Security as soon as possible.
• Return checks to Social Security as soon as possible. DO NOT CASH any checks received for the month in which the beneficiary died or thereafter. For example, if the person dies in July, you must return the benefit paid in August.

A one-time payment of $255 can be paid to the surviving spouse if he or she was living with the deceased; or, if living apart, was receiving certain Social Security benefits on the deceased's record. If there is no surviving spouse, the payment is made to a child who is eligible for benefits on the deceased's record in the month of death.

Some of the deceased's family members may be able to receive Social Security benefits if the deceased person worked long enough under Social Security to qualify for benefits. You should get in touch with your local Social Security office as soon as you can to make sure the family receives all of the benefits to which they may be entitled.

Social Security uses the deceased worker’s basic benefit amount and calculates what percentage survivors are entitled to. The percentage depends on the survivors’ ages and relationship to the worker. Here are the most typical situations:

• A widow or widower, at full retirement age or older, generally receives 100 percent
• A widow or widower, age 60 or older, but under full retirement age, receives about 71-99
• A widow or widower, any age, with a child under age 16, receives 75 percent of the worker’s benefit amount.
• Children receive 75 percent of the worker’s benefit amount.

You may also find these links useful :
http://socialsecurityhop.com/comment/834/ss-benefits-widows
http://www.ssa.gov/pubs/10084.html#3

I was browsing on your website, but I can not find the section where I can download the application form for benefits. Please help!!!

I'm not sure what benefit you're looking to apply for (retirement, disability, survivors etc.), but here's a good place to start :

http://www.ssa.gov/onlineservices/

We have no direct experience with this. The following web site states the following :

You can’t sue just because Social Security has turned you down once or twice. You must have been turned down by a Social Security Administrative Law Judge and by Social Security’s Appeals Council. You may be three years into your case before this happens.

The site has other details that may be of interest.

http://www.charleshallfirm.com/im-so-mad-can-i-sue-social-security/

You can apply for Social Security retirement benefits in one of three ways :

• Online - use the Social Security Retirement Benefit Application.
• By phone - call the SSA at 1-800-772-1213. If you are deaf or hard of hearing, you can call at TTY 1-800-325-0778.
• In person - visit your local Social Security office. (Call first to make an appointment.)

You may need to bring documents such as your Social Security card (or a record of your number), your original birth certificate, and your most recent W-2.

http://www.socialsecurity.gov/retire2/applying8.htm

http://www.socialsecurity.gov/retire2/applying5.htm

I recommend you contact the SSA directly for this. Also, you may want to contact legal counsel. It appears the definition of "marriage" varies from state to state, and so your status as to benefits may vary depending on your state.

However, Social Security documents indicate that even if you are "separated", as long as you have not obtained a divorce decree the SSA considers you married for purposes of Social Security benefits.

For example, see the following case where even though a husband and wife were not living together at the time of the husband's death, they were still married. The SSA seems to want "evidence" (ex. divorce decree) of a divorce.

https://secure.ssa.gov/poms.nsf/lnx/1506210042

However, even if you are deemed "divorced", you likely still will qualify for benefits based on his work record.

http://ssa-custhelp.ssa.gov/app/answers/detail/a_id/299