What are the common grounds for relief from Deportation?

The initiation of removal proceedings against you does not necessarily mean that you will be deported from the United States. There are many forms of discretionary relief from deportation that may be available to you, depending on the facts of your case.
It is up to the discretion of the immigration judge to grant your requested relief. So even if you meet all of the criteria for a particular relief, the judge could still deny that relief. If the immigration judge does grant relief, the removal proceedings are stopped or postponed, and in some cases you are then granted lawful permanent residence.
Below is an overview of the more common forms of relief.

1.    Voluntary Departure

In some cases, a person facing removal proceedings may be permitted to depart from the United States voluntarily, at their own expense. An Immigration and Customs Enforcement official may permit you to do so prior to your removal hearing. During your removal hearing, an immigration judge may permit voluntary departure only if you admit your removability. To be eligible, you must be able to show that you can pay your departure expenses (except in special circumstances), and the charges against cannot be for an aggravated felony or terrorism.
You may also be permitted to depart voluntarily at the conclusion of your removal proceedings if you (1) have been physically present in the United States for at least one year before the Notice to Appear was served on you, (2) you have been a person of good moral character for the previous five-year period, (3) you are not removable for aggravated felony or removal grounds, and (4) you can show that you have the means to and intend to depart from the United States.
Voluntary Departure is beneficial because it is technically not considered a removal, and you would not be subject to the ten year bar on returning to the United States that usually comes with a removal order.
 
2.    Cancellation of Removal

Cancellation of removal is available for permanent residents if you meet the following requirements:
(1)    You have been lawfully admitted for permanent residence for not less than five years
(2)    You have resided in the United States continuously for seven years after having been admitted in any status, and
(3)    You have not been convicted of any aggravated felonies.
It is also available for non-permanent residents, or permanent residents who cannot meet the requirements set forth above, if the following requirements can be met:
(1)    You have been physically present in the United States for a continuous period of not less than ten years,
(2)    You have been a person of good moral character,
(3)    You have not been convicted of any of the crimes or offenses that would make him or her inadmissible or removable,
(4)    You are not subject to any of the security grounds of inadmissibility or removability, and
(5)    You can show that removal would result in exceptional and extremely unusual hardship to your citizen or lawful permanent resident spouse, parent, or child.
Note that these requirements may be reduced for victims of abuse or domestic violence.
If you are not a permanent resident and you qualify for cancellation of removal, then the immigration judge will adjust your status to permanent resident as of the date of the cancellation of removal.

3.    Adjustment of Status

If you qualify for adjustment of status, you can be spared having to go abroad to wait for your application for adjustment to be approved, which can take several months to several years. To be eligible for adjustment of status you must file an application for adjustment and meet the following requirements:
(1)    You were inspected and admitted to the United States (you did not enter illegally);
(2)    You are eligible for a permanent resident visa;
(3)    You are admissible; and
(4)    An immigrant visa is immediately available.
4.    Waivers
Depending on the grounds upon which your removal proceeding is based, you may be able to apply for a waiver. Waivers usually require you to show that your removal would result in extreme hardship to your legal permanent resident or citizen spouse, children, or parents. Two common examples of waiver of inadmissibility are those for aggravated felonies under INA section 212(h) and for fraud or misrepresentation under INA section 212(i). Usually waivers of inadmissibility are applied for in conjunction with an application for adjustment of status, described above.

5.    Asylum

Asylum is available to anyone, despite their immigration status, who meets the requirements of refugee status. A refugee is defined basically as any person who is outside of their country and “is unable or unwilling to return to, and is unable or unwilling to avail himself or herself of the protection of, that country because of persecution or a well-founded fear of persecution on account of race, religion, nationality, membership in a particular social group, or political opinion.”
This form of relief is not available to persons who have participated in the persecution of others, those who have committed particularly serious crimes, and those who present a danger to the United States. Generally, you must apply for asylum within one year of entering the United States, unless you can show that changed circumstances materially affected your eligibility for asylum or that extraordinary circumstances caused you to delay filing your application.
If you are granted asylum, your status as asylee continues indefinitely, until circumstances change in your country so that you no longer qualify as a refugee.

6.    Stay of Removal

If you are given a final order of removal, you may apply through CIS (Citizenship and Immigration Services) for a stay of that order. If granted, your removal will be put on hold for whatever duration of time CIS approves. While not a true form of relief, in that it is only temporary, getting a stay of removal will give you time to appeal your case or apply for immigrant status.

7.    Registry

You may qualify for registry, and may thus be eligible for permanent resident status, if you entered the United States before January 1, 1972 (even if you entered without inspection), you have been a continuous resident of the United States, you are a person of good moral character, and you are otherwise eligible for citizenship