Rabbi Dov Lior of Kiryat Arba made a ruling "in principle" saying that modern stainless steel cookware does not absorb flavor at a halachiclly significant level.
Apparently two Avrachim from the Torat HaChaim yeshiva tested the amount of absorption of stainless steel and found it to be one part in 170,000 of the volume of the fluid cooked in the pot. In response to this test Rav Lior wrote:
"זוהי בליעה מזערית ביותר, ולכן נראה בפשטות שכלי נירוסטה הוא ככלי
This is an extremely insignificant amount of absorption, and thus it
seems plainly that stainless steel pots are like glass.
Further, a letter was later published in the Makor Rishon newspaper in which he says:
"בשעתו אמרתי שאם יתברר בוודאות שיש חומר כלשהו שאינו בולע, דינו יהיה
ככלי זכוכית, שלדעת ה'שולחן ערוך' אינו בולע ולכן אינו טעון הגעלה לפסח
At the time I said that if it will become certain that there is any
substance that does not absorb, his law will be like the law of glass,
that according to the Shulchan Aruch does not absorb and thus does not
require "hagala" for Pesach and it is sufficient to wash it.
Rav Lior concludes:
"אף לדעת הרמ"א, המחמיר בכלי זכוכית, הדברים אמורים דווקא בכלי זכוכית
שתחילת יצירתם מחול ולכם דינם כחרס. אולם כלי נירוסטה, שאינם בולעים כמעט
כלום ונעשו ממתכת ולא מחול, מותר לבשל בהם בשר, לנקות היטב ואחר כך לבשל
חלב באותו היום, והן להפך".
Even according to the opinion of the Rama, who is stringent with glass
utensils, saying that since they are initially made from sand their
ruling is like earthenware. But stainless steel utensils, that don't
hardly absorb anything and are made of metal not sand, it is
permissible to cook in them meat, to clean well and after to cook milk
the same day, and the reverse as well.
In his letter to the two avrechim he says:
"מכיוון שדין זה עלול להתפרש כדבר חדש, בקשתי היא שיסכימו איתי עוד שני
פוסקים ויצטרפו להיתר זה"
Since this ruling might be interpreted as something new, I request
that two other poskim will agree with me.
All this according to Rav Ratzon Arusi.
Here is a video in which Rav Bar Chaim from Machon Shilo discusses his support of Rav Lior's ruling and his belief that it would also apply to forbidden substances. He mentions the above reported by Rav Arusi and also discusses a conversation with a Rav Dr. Moshe Antelman a PhD chemist who worked in industrial chemical engineering who joked with him that the absorption was, "approximately three atoms". In seriousness he said it was clear to him that there is no halachiclly significant absorption.
Rav Bar Chaim goes on to point discuss the 1/60th rule and how there was a machloket whether or not it applied only to earthenware pots or if it also applied to metal pots. He brings Rambam הלכות מאכלות אסורות יז to support that this is only regarding earthenware and points out that Ra'avad agrees with Rambam in this case as does the Rashbah and the Ran and list the sources.
קדירה של חרס שנתבשל בה בשר נבילה, או בשר שקצים ורמשים--לא יבשל בה בשר
שחוטה באותו היום. ואם בישל בה מין בשר, התבשיל אסור; בישל בה מין אחר,
בנותן טעם. [ב] ולא אסרה תורה אלא קדירה בת יומה בלבד, הואיל ועדיין לא
נפגם השומן שנבלע בקדירה; ומדברי סופרים, לא יבשל בה לעולם. לפיכך אין
לוקחין כלי חרס ישנים מן הגויים שנשתמשו בהן בחמין, כגון קדירות וקערות,
לעולם, ואפילו היו שוועין באבר; ואם לקח, ובישל בהן מיום שני
A pot of earthenware in which the meat of a nevilah ( non-kosher meat ) was cooked, or the
meat of bugs or insects -- don't cook in it kosher meat on the same
day. And if he cooked the same type of meat in it, the cooked dish is
forbidden; if he cooked a different type of meat then it is according
to if it improved the flavor. And the Torah only forbid a pot on the
same day, because the fat that has been absorbed in the [ earthenware
] pot has not yet gone bad; and our Sages forbid cooking in this pot
forever. Thus, we do not take old earthenware pots from gentiles who
have cooked hot in them, such as pots and bowls, ever, and even if it
was [ something I can't translate ]; and if one took, and cooked in
it from the second day on -- the cooked food is permissible.
He points out that the majority of Rishonim agree with the statement from Avot that glass does not absorb and that he agrees with Rav Lior's statement that stainless steel is like glass and that regular cleaning with soap is sufficient, i.e. that hagalah is not required. He further points out that according to his opinion washing with soap causes any matter that might remain on the surface of the pot to become "pagum", no longer considered food.
A side note of interest is that according Rambam's method, in specific cases where transfer of flavor is the criteria used ( and it's not used in all cases ) then it only applies if there is a trusted gentile physically present to taste the food for you and tell you if flavor has been transferred. Otherwise, transfer of taste is not applicable and you go by the 1/60 rule ( or 1/100 or 1/200 depending on the particular substance involved ). Another thing to point out is that transfer of flavor doesn't apply if the flavor transferred is a unpleasant or bad flavor. For example if a mouse where to be found in a barrel of wine it wouldn't make if forbidden because it doesn't improve the flavor of the wine.
Here's a list of the properties of stainless steel on wikipedia.