What level of kedushah does the Tetragrammaton-substitution "yod-yod" have? Can it go in the garbage? Must it be geniza'ed? Let's assume that the word is used in a song or other non-Torah sentence.
(related: Double Yud as G-ds Name)
Just recently (22 Cheshvan, 5774) a letter from the Lubavitcher Rebbe discussing the Yud-Yud spelling was published in a leaflet - page 22. In that letter he mentions the opinion of Rabbi Yaakov Meir Garadenski from Minsk that a siddur should be specifically printed with the Yud-Yud, and that the source siddur for the printing of Tehilas Hashem followed this opinion. That opinion holds that it makes it less problematic when children mishandle the siddur. (The letter continues that Chabad tradition does not hold that way - as in it is not concerned about the mishandling by children, just that there was no time to change it for that printing. Alex discussed some that history here).
So from this we see it has a lower level of kedusha. However, it is a letter of Hashem's name which is referencing Hashem. The discussion of the requirements to treat such a usage respectfully in a non-sefer is discussed around the preference of using ב"ה vs. בס"ד. It would seem that that in this case, the same discussion applies (basically it is a Machlokes Achronim). Specifically in the case of the two yuds, there is more place to be stringent (thanks @fred).