It is a Machlokes if a person admits partially to a claim and immediately offers the amount that he admitted if he is Chayav a Shevu'as Modeh b'Miktzas.
If someone claims that his friend owes him utensils and Karka and he admits that he owes utensils but he denies that he owes Karka he is Patur from a Shevu'ah. (1)
Rami Bar Chama says the four Shomrim are not Chayav a Shevu'ah unless they are Modeh b'Miktzas. (2)
If a person denies a claim completely and witnesses testify that he is partially Chayav it is a Machlokes if he is Chayav a Shevu'as Modeh b'Miktzas.
If someone claims that his friend owes him wheat and the friend admits that he owes him barley according to the Tana Kama he is Patur from a Shevu'ah, while R. Gamliel disagrees.
If a Gazlan is Chayav a Shevu'as Modeh b'Miktzas the Rabanan decreed that the person who is making the claim swears instead of him since he is suspected to swear falsely. (3)
If someone denies a claim completely he is Chayav a Shevu'ah Heses d'Rabanan. (4)
A shepherd who takes his own animals out to graze is Pasul for a Shevu'ah because most shepherds allows their animal to graze in private property and a Gazlan is Pasul for a Shevu'ah. (5)
There is a Chazakah that a person doesn't sin unless he will profit from the sin..
If two people are holding on to a Talis and each one claims that it belongs to him each one shall swear that he has a share in the Talis and that his share is not less than half.
R. Yochanan says that the reason for the Shevu'ah is a Takanas Chachamim so that a person will not grab his friend's Talis and claim that it belongs to him.
R. Yochanan holds that a person who is suspected to steal money is not suspected to swear falsely. (6)
If someone denies a loan he is Kosher for Eidus
If someone denies a Pikadon he is Pasul for Eidus if witnesses testify that at the time of his denial the Pikadon was in his possession.
If a Shomer is Patur from paying but he chooses to pay instead of taking a Shevu'ah he must swear that the Pikadon is not in his possession. (7)
People have the perception that the prohibition of Lo Sachmod only applies if you are not willing to pay for the object of your desire.
A BIT MORE
1. Because there is no Chiyuv Shevu'ah on Karka.
2. A Shomer is only Chayav a Shevu'ah if the owner claims that he gave him three animals to watch and they all died as a result of negligence and the Shomer denies one completely and says that he was only given two animals and one he admits that it died as a result of negligence and regarding the third he says that it died b'Ones. The Shomer must swear a Shevu'as ha'Shomrim that the animal died b'Ones along with a Shevu'as Modeh b'Miktzas.
3. However, if he is Chayav a Shevu'ah d'Rabanan since the Rabanan do not make one Takanah on top of another Takanah and therefore if the person who is Chayav the Shevu'ah is a Gazlan no Shevu'ah is made at all.
4. The Rabanan are Mechayev him a Shevu'ah because there is a Chazakah that a person doesn't make a baseless claim.
5. However, a shepherd who grazes the animals of other people is not Pasul since he won't allow them to graze in private property because a person doesn't sin unless he stands to profit from his sin.
6. Because people have the perception is that the prohibition of a Shevu'ah is more stringent than the prohibition of Gezel.
7. Even though he is paying for the item he must swear that it is not on his possession because we are concerned for the possibility that he is paying for it because he wants to keep the item for himself.