Gemara

Had it said only "It will not come upon you", one would think that even a clothing salesman may not put on Sha'atnez (to display it);

"Do not wear" teaches that it is forbidden only to wear it like clothing, i.e. when he benefits from it.

Had it said only "Do not wear", one might have thought that only wearing is prohibited, since there is great benefit;

"It will not come upon you" forbids putting it on yourself.

Beitzah 14b - Mishnah: One may send Kelim on Yom Tov, even if they contain Sha'atnez, as long as they are needed for Yom Tov.

Question: What use is there for Kelim with Sha'atnez?

Suggestion: One may fold it under himself (like a cushion).

Rejection (Beraisa): "Lo Ya'aleh Alecha" - but (mid'Oraisa) one may spread Sha'atnez under himself;

However, Chachamim forbade this, lest a strand of the Sha'atnez bend and come upon his flesh.

Suggestion: Perhaps Chachamim permit when a garment is between the person and the Sha'atnez!

Rejection: R. Shimon ben Pazi forbids to sleep over Sha'atnez even if ten sheets are in between.

Answer #1: One may send a curtain with Sha'atnez.

Objection: But Ula taught that a curtain is Mekabel Tum'ah because a servant drapes himself in it for warmth!

Answer #2: Rather, one may send stiff garments with Sha'atnez.

Rav Huna brei d'Rav Yehoshua: Stiff felt garments (of Sha'atnez) from Narash (a certain place) are permitted.

Rava: A money pouch with Sha'atnez is permitted (one may put it under his garment, for the coins make it stiff and it does not give warmth). A seed bag with Sha'atnez is forbidden.

Rav Ashi: Even a seed bag is permitted, for this is not the normal way to get warmth.

Tamid 25b (Mishnah): Young Kohanim would fold the Bigdei Kehunah and put them under their heads (when they slept).

Question: May one benefit from Bigdei Kehunah?!

Answer: It means, they would put them near their heads.

Support: Presumably, this is correct. According to the opinion that a regular Kohen's Avnet was (Sha'atnez) like that of the Kohen Gadol, we cannot say that they slept on them, for Kil'ayim is forbidden (except for needs of Avodah)!

Mid'Rabanan, one may not spread Sha'atnez under himself. One may not sit over Sha'atnez, even if ten mattresses are in between.

Rejection #1: Perhaps the other Bigdei Kehunah (without Sha'atnez) were under their heads!

Rejection #2 (Rav Ashi): Bigdei Kehunah are stiff;

Rav Huna brei d'Rav Yehoshua: Stiff felt garments from Narash are permitted.

Rishonim

The Rif and Rosh (Beitzah 7a and 1:25) bring the Gemara and Rav Ashi's conclusion.

Question (Ran DH Gemara): The Gemara forbids sitting on Sha'atnez 'lest a thread wrap itself over him', which connotes that there is nothing in between. This is forbidden even when something is in between, lest one put the Sha'atnez on himself!

Answer #1 (Rashi): When nothing separates, the concern lest a thread wrap itself over him is mid'Oraisa. When something is in between, it is forbidden mid'Rabanan because of the same concern.

Answer #2 (Ran and Ri Korkus Kil'ayim 10:12): When nothing separates we forbid (lest a thread wrap...) any Sha'atnez. When something is in between, the only concern is lest he put the Sha'atnez on himself. This applies only to garments normally put on a person.

Ri Korkus (ibid.): The Rambam explains that the decree is solely lest a thread wrap itself over him. It makes no difference whether or not anything separates.

Hagahos Ashri: One may spread stiff Sha'atnez under himself, since even soft Sha'atnez under himself is forbidden only mid'Rabanan. Chachamim did not decree about stiff Sha'atnez. One may not wear stiff Sha'atnez.

Rambam (Hilchos Kil'ayim 10:12): One may make and sell garments with Sha'atnez. It is forbidden only to wear them or cover oneself with them - "Lo Ya'aleh Alecha".

Source: Mishnah Kil'ayim 8:1.

Rambam (ibid.): One may sit under an Ohel of Sha'atnez. Similarly, the Torah permits to sit on sheets of Sha'atnez - "Lo Ya'aleh Alecha", but you may spread it under yourself. Mid'Rabanan it is forbidden to sit on Sha'atnez, even if the bottom of ten sheets is Sha'atnez, lest a thread wrap itself over his skin.

Rambam (13): This (Isur to sit on Sha'atnez) applies to soft things, like curtains and garments. One may sit or lie on stiff things that do not wrap, such as pillows and blankets, if they do not touch his skin.

Ra'avad: The Yerushalmi permits only if they are empty, and not on a bed (only on a stiff ledge or piece of wood).

Kesef Mishneh: The Rambam did not need to say this, for he forbids soft things.

Gra (YD 301:5): The Rambam omitted it, because the Bavli does not distinguish. Tosfos (Shabbos 57b DH Ein) holds that only regarding Kil'ayim mid'Rabanan we do not forbid if it is stiff.

Rosh (Hilchos Kil'ai Begadim 8, in Perek 9 of Hilchos Nidah): Sha'atnez does not apply to pillows and blankets because they are made to lie on. "Lo Ya'aleh Alecha", but you may spread it under yourself. Chachamim forbade to do so, lest a thread wrap itself over his skin. This is even if the bottom of ten sheets is Sha'atnez. Pillows and blankets are stiff and they do not normally wrap on his skin, so they are permitted if they do not touch his skin. The Yerushalmi (Kil'ayim 9:1) permits only if they are empty and on a ledge or piece of wood. If they are full, or on a bed or straw it is forbidden, for they bend under him and wrap on his skin.

Rosh (Tamid 28a DH Kashin): Every strand of Bigdei Kehunah is six-fold. Chachamim did not forbid stiff Sha'atnez under a person. We are not concerned lest a strand bend.

Mefaresh (27b DH Rav Ashi): Bigdei Kehunah are stiff so they do not give benefit (warmth), so they are permitted even above a person.

Poskim

Shulchan Aruch (YD 301:1): The Torah permits to sit on sheets of Sha'atnez. "Lo Ya'aleh Alecha", but you may spread them under yourself.

Rema: Similarly, one may put them over himself in a way that does not warm him, e.g. to spread an Ohel of Sha'atnez over himself, or over a bucket of hot water to keep it hot. It is forbidden only to wear them or to put them on in a way that gives warmth.

Source (Gra 2): The Gemara forbids curtains of Sha'atnez because a servant drapes himself in it for warmth. Otherwise, it would be permitted, even though they preserve heat (in a room) and protect from the sun and rain. The Mishnayos (Kil'ayim 9:5-6) say that people who sell or sew Kil'ayim may not intend for benefit, and Tzenu'im put the Kil'ayim on a pole or the ground. This is because the Isur is to wear it or put in on his body.

Taz (2): The Rema and Tur mention an Ohel amidst permitted uses of Sha'atnez mid'Oraisa, implying that mid'Rabanan it is forbidden. However, the Rema teaches Ohel together with covering a bucket, which is permitted even mid'Rabanan. The Tur later forbids mid'Rabanan sitting over sheets. We infer that Chachamim did not forbid the other things, such as an Ohel. However, it is not clear why Sha'atnez curtains on the sides of wagons are permitted. Standard curtains are forbidden because the servant wraps and warms himself in it. The same applies in wagons; people are often pressed against the sides! Indeed, some are stringent not to sit in such wagons for this reason. Most merely try to avoid touching them. Even though this is difficult, they are lenient because it is not Sha'atnez mid'Oraisa, for it is not Shu'a, Tavuy and Nuz together. Even if they touch it, it is Davar she'Eino Miskaven. One should be as careful as possible not to touch.

Shulchan Aruch (2): Mid'Rabanan, even if the bottom of ten sheets is Sha'atnez, one may not sit on them lest a thread wrap itself over his skin. This refers to soft Sha'atnez, such as curtains and garments. Regarding stiff pillows and blankets we are not concerned lest a thread wrap itself over his skin. One may sit over them if they do not touch his skin.

Gra (4): Other Meforshim are not concerned whether or not they are stiff. It depends only on whether there is concern lest they wrap on his skin.

Tif'eres Yisrael (Batei Kil'ayim (after Zera'im) 129): 'Kashim' does not mean hard like a rock. Rather, it means stiff.

Shulchan Aruch (ibid.): This is if they are empty and on a ledge or piece of wood. If they are full, or on a bed or straw it is forbidden, for they bend under him and wrap on his skin.

Rema: Some say that this refers to Sha'atnez mid'Oraisa, but one may sit on Sha'atnez mid'Rabanan that is stiff in any case. Since when they are soft it is only mid'Rabanan, we do not decree when they are stiff. Nevertheless, l'Chatchilah one should not make them from Sha'atnez.

Shach (2) and Gra (5): All agree to this! One may even wear stiff Kil'ayim mid'Rabanan (that was not Shu'a, Tavuy and Nuz together), since even if it was soft it would be permitted mid'Oraisa. The Rema himself permits this regarding felt (Sa'if 2). The stringent opinion holds that felt is Kil'ayim mid'Oraisa.

Gra (6): L'Chatchilah one should not make them from Sha'atnez lest someone remove the inside and cover himself with it.