If a utensil was used for many different methods of cooking and storing food, there is a dispute among the Rishonim concerning what method of koshering should be used. The Raavad (ע”ז עו. ד”ה רב אשי) and Raavya (סימן תסד) rule that if a utensil was mostly used for cooking and it was just used a few times for roasting, the utensil needs libun. This is due to the fact that there are some beliot in the pot that got into the pot through direct heat and those beliot will not come out without libun. Rabbeinu Tam (תוס’ ע”ז עו. ד”ה אמר) also rules that we concern ourselves with the lesser usage of the utensil, provided that one uses the utensil this way consistently, and it wasn’t a one-time use in an abnormal way. Therefore, a cast iron pot that one sometimes uses for cooking and only occasionally uses for roasting, according to Rabbeinu Tam would require libun, even though the primary usage of the pot is for cooking. The Ritva (פסחים ל: ד”ה הני) adds that if one uses a utensil only one way, and is unsure if they ever used the utensil the other way, there is no requirement to be stringent by koshering it with a higher level of koshering on the chance it was used a different way.
The Ramban (ע”ז שם ד”ה לעולם) rules that we always treat a utensil based on the most common usage. Therefore, if one usually used a utensil for cooking and a few times used it for roasting, the utensil only needs hagala. If one stores food in a pot, and only occasionally uses the pot for cooking, one would only be required to wash the pot well to kosher it, hagala is not required.
The Shulchan Aruch (או”ח תנא:ו) rules the same as the Raavad, that we are only concerned with the general usage of a utensil. The Rama rules that one should be stringent, the same as the Ramban’s ruling. However the Rama is lenient. He ruled that if one already koshered the utensil and used it, the food is permitted.
Rov tashmisho means the general use of the utensil in regard to the beliot you are koshering the utensil to get rid of. If a bowl was used for cold kosher food all the time, and was used for hot non-kosher soup one time, the bowl would need hagala according to all opinions. This is because in regard to prohibited beliot, the utensil was only used for cooking. However, if a bowl is used for cold chametz all the time, and is used for hot chametz once, then it would only need to be washed well before using it on Pesach, according to those that rely on rov tashmisho.