What may and may not be moved on Shabbat, by category, with Rabbi Shay Tahan’s rulings from this site. Sources are Shulchan Aruch Orach Chaim 308–311 and the Mishnah Berurah; the siman and se’if are given on each line.
Quoted text is Rabbi Tahan’s own wording from the linked answer; lines without a link follow the Shulchan Aruch source given, as confirmed by Rabbi Tahan.
What it is: A utensil or object whose main use is permitted on Shabbat — a chair, a cup, a pillow, clothing.
May it be moved? May be moved to use it, for its place, and even just to protect it from breaking or being stolen. It may not be moved for no purpose at all. Food and holy books may be moved even for no purpose.
Source: Shulchan Aruch O.C. 308:4

Chair, table, cup, clothingRabbi Tahan’s ruling
Standard examples of a kli shemelachto l’heter — may be moved for any need (Shulchan Aruch O.C. 308:4).

Vase of flowersRabbi Tahan’s ruling
May be carried from the dining-room table to a coffee table on Shabbat — “Yes.”

House keysRabbi Tahan’s ruling
Apartment keys may be carried in your pocket on Shabbat where there is an eruv — “Yes.” An electronic car key is different: “It’s muktse. You shouldn’t carry muktse on Shabbat.”
Source: Carrying Keys on Shabbat · Rabbi Tahan, WhatsApp Q&A, Jul 16 2019 (car key)

Clothes in the dryerRabbi Tahan’s ruling
Clothes that were fully dry before Shabbat may be taken out of the dryer on Shabbat — “Yes.”

Bed in the sukkahRabbi Tahan’s ruling
A bed brought into the sukkah may be moved out again to make room for the table — “Yes, you may move it.”
Source: Moving a Bed in the Sukkah

Snow shovelRabbi Tahan’s ruling
“The shovel is also permitted to be used since it’s a special shovel for snow and isn’t muktza, and even if it was muktza, it would be kli shemelachto l’issur that is permitted to use (l’tzorech gufo).”
Source: Q&A: The Halachot of Snow

Lego, games and puzzlesRabbi Tahan’s ruling
Lego: “Yes” (Lego on Shabbat). On games for Sephardim: “We hold that games are muktse, but if you must play there are some leniencies. Lego is a machloket. Backgammon and chess is fine if you don’t violate borer. Puzzles, some allow if it doesn’t have a frame.” “Must play” = e.g. kids who very much need you to play with them, or someone who needs to clear his mind.
Source: Lego on Shabbat · Playing Games On Shabbat · Rabbi Tahan, WhatsApp Q&A, May 14 2020 — class on board games
What it is: A utensil whose main use is a melacha — a hammer, scissors, a pen, a sponge.
May it be moved? May be moved to use it for something permitted (l’tzorech gufo — e.g. a hammer to crack nuts) or because its place is needed (l’tzorech mekomo). It may not be moved only to protect it from the sun, from breaking or from theft. Touching it without moving it is permitted (Rama).
Source: Shulchan Aruch O.C. 308:3

Hammer (to crack nuts)Rabbi Tahan’s ruling
The Shulchan Aruch’s own example: a hammer may be moved to crack nuts with it, or when its place is needed (O.C. 308:3).

Sponge in the sinkRabbi Tahan’s ruling
“M.B. allows to remove the sponge, but since it’s כלי שמלאכתו לאיסור you can only move it if you need the space.”

Scissors to open a food packageRabbi Tahan’s ruling
“Yes, but it’s better to rip with your hands.”
Source: Opening Packages on Shabbat
What it is: A valuable or delicate item that one is careful not to use for anything but its special purpose — the Shulchan Aruch’s examples: a shochet’s knife, a mohel’s knife, a barber’s scissors, a scribe’s quill knife.
May it be moved? May not be moved at all — not even for its place or for a permitted use.
Source: Shulchan Aruch O.C. 308:1

Barber’s scissors, shochet’s knife, mohel’s knifeRabbi Tahan’s ruling
The Shulchan Aruch’s examples of muktzeh machmat chisaron kis — may not be moved on Shabbat (O.C. 308:1).
What it is: Something that is neither a utensil nor food and has no Shabbat use — stones, sticks, sand, money, animals, raw inedible items.
May it be moved? May not be moved at all, even for a permitted use and even if its place is needed (Rama). Animals and birds may not be handled.
Source: Shulchan Aruch O.C. 308:7 · Shulchan Aruch O.C. 308:39

Stones and pebblesRabbi Tahan’s ruling
“Something that has nothing to do with utensils, for example pebbles and stones, even though they are suitable for covering a utensil, are forbidden to move” (O.C. 308:7).

Animals and petsRabbi Tahan’s ruling
“Are animals muktzeh on Shabbat? — Yes. Even if it’s my dog? — Yes. (R’ Auerbach and R’ Moshe had a leniency on pets.)” Touching pets: “Muktse, according to most opinions.”
Source: Animals Muktze · Touching Pets on Shabbat · Class: Are Pets Muktza?

InsectsRabbi Tahan’s ruling
“One is not even allowed to move these insects, as they are considered muktzeh.” A dead insect that causes discomfort is a graf shel re’i and may be removed.

Magnetic / credit cardRabbi Tahan’s ruling
“The magnetic card is muktze, and one may not carry it on Shabbat.”

Raw or frozen meatRabbi Tahan’s ruling
“The Shulchan Aruch says not muktza because it’s possible to eat. However, some poskim say that today you can’t and won’t eat raw meat, therefore it is muktza (R’ Elyashiv). Raw chicken would have the same halacha. Raw fish depends on if that type of fish is eaten raw, for example salmon in sushi.” Taking meat out of the freezer on Shabbat to defrost for after Shabbat: “Machloket. You can be lenient if you can’t take out before.”
Source: Is Frozen Meat Muktza · Rabbi Tahan, WhatsApp Q&A, Apr 2 2021 (defrosting) · Shulchan Aruch O.C. 310:2 (edible food is not muktzeh)
What it is: An item that was serving a forbidden use at the start of Shabbat — the classic case is a lamp that was burning at twilight.
May it be moved? A lamp that was lit when Shabbat began may not be moved the whole Shabbat, even after it goes out.
Source: Shulchan Aruch O.C. 279:1 · Shulchan Aruch O.C. 308:3

Lamp that is onRabbi Tahan’s ruling
Moving a lamp on Shabbat: “No.” (Asked together with a fan and a heater that were turned on before Shabbat: “Fan yes, heater and lamp no.”)

Space heater / radiator that is onRabbi Tahan’s ruling
“These heaters are muktzeh on Shabbat and may not be moved for any reason, unless there is a danger of fire or another safety issue. The reason they are muktzeh is that the heating elements inside the radiator become red hot and are considered fire, while the radiator itself serves as the base (בסיס) for that fire.”
Source: Cold Weather and Shabbat: A Practical Halachic Guide to Home Heating · Moving Electric Appliances on Shabbat

Appliance switched off by a timerRabbi Tahan’s ruling
Unplugging it while the timer is still in the wall: “No. A. It’s muktse. B. Some would hold since it’s connected to the wall it’s melacha of סותר.”
What it is: A permitted item on which a muktzeh item was resting when Shabbat began becomes muktzeh itself — a bed with money on it.
May it be moved? A bed that had money on it at twilight may not be moved (even for its place or a permitted use — Rama). If it holds both a muktzeh item and a more important permitted item, it may be moved. If the muktzeh item was left there by mistake, the base may be tilted so it falls off.
Source: Shulchan Aruch O.C. 310:7 · Shulchan Aruch O.C. 310:8 · Shulchan Aruch O.C. 309:4

Fridge or freezer door with a muktzeh item on itRabbi Tahan’s ruling
The door does not become a base: the item was put there only for lack of space (M.B. 309:19); it is insignificant relative to the door (M.B. 310:31, 277:7 — R’ Shlomo Zalman Auerbach and R’ Elyashiv extend this to a fridge door); and other food on the door makes it a base for permitted items too (O.C. 310:8). “One can safely assume that the freezer door is not a base to any muktze items in it, and therefore one can open the door on Shabbat.”
Source: Opening Freezer on Shabbat

A few coins in a drawerRabbi Tahan’s ruling
“A few pennies left in a drawer do not make the entire drawer a base, since they are relatively insignificant (M.B. 310:31).”
Source: Opening Freezer on Shabbat
What it is: Something new that did not exist, or was not usable, when Shabbat began.
May it be moved? An egg laid on Shabbat may not even be touched; one may cover it with a utensil so it does not break.
Source: Shulchan Aruch O.C. 322:1

Egg laid on ShabbatRabbi Tahan’s ruling
The Shulchan Aruch’s case of nolad — forbidden even to touch (O.C. 322:1).

Snow and rainRabbi Tahan’s ruling
“Snow isn’t muktza. The Gemara says that rain water isn’t muktza, and this is also how the Shulchan Aruch rules (סי’ שלח ס”ח). The poskim compare snow to rain and say it was already in the clouds and therefore isn’t muktza.” Snow itself is not muktzeh, but making snowballs or crushing it into water is a problem (confirmed on WhatsApp).
Source: Q&A: The Halachot of Snow · Rabbi Tahan, WhatsApp Q&A, Jan 3 2021

Whipped cream from a spray canRabbi Tahan’s ruling
“It’s permitted; however, the Shmirat Shabbat is stringent.”
Source: Whipped Cream on Shabbat
What it is: Items dedicated to a mitzvah, such as sukkah decorations.
May it be moved? Food and decorations hung in the sukkah to beautify it may not be used all eight days, and on Shabbat and Yom Tov may not be moved, because they are muktzeh.
Source: Shulchan Aruch O.C. 638:2

Sukkah decoration that fellRabbi Tahan’s ruling
Re-attaching a paper decoration that fell from the sukkah roof on Shabbat or Yom Tov: “No, it’s muktza.”

ShofarRabbi Tahan’s ruling
“It is forbidden to move a shofar unless for the sake of using it or for its place” (Rama, O.C. 308:4).
What it is: Everyday electric items asked about on this site.
May it be moved? See each line — these are Rabbi Tahan’s answers.

Fan running from before ShabbatRabbi Tahan’s ruling
May be moved without unplugging — “Fan yes” (heater and lamp: no).

Cell phone used only as an alarm clockRabbi Tahan’s ruling
“If it’s only used for alarm and nothing else it’s fine” to move it; put it on airplane mode so no signals are triggered.
Source: Moving a Cell Phone on Shabbat
Touching without movingRabbi Tahan’s ruling
“It is only forbidden to move muktzeh, but touching without moving is permitted” (Rama, O.C. 308:3).
Moving muktzeh with two peopleRabbi Tahan’s ruling
Is one allowed to move a muktzeh item on Shabbat if it’s moved by two people? — “No.”
Source: Moving Muktze
Indirect moving (טלטול מן הצד)Rabbi Tahan’s ruling
“Indirect movement of a muktze item is permitted if one is not moving it for the sake of the muktze item, rather for the sake of a permitted item (S.A. O.C. 311:8)” — e.g. opening a freezer door that has a muktzeh item on it.
Source: Opening Freezer on Shabbat · Shulchan Aruch O.C. 311:8
Moving with one’s body (טלטול בגופו)Rabbi Tahan’s ruling
“Moving muktzeh with one’s body is permitted even when moved for the sake of the muktzeh” (Rama, O.C. 311:8).
Food is not muktzehRabbi Tahan’s ruling
“No food item which is edible and picked is muktzeh on Shabbat” (O.C. 310:2). Raw meat today: see the meat line above.
Related — hachana (preparing for after Shabbat)Rabbi Tahan’s ruling
Not a muktzeh rule, but asked together: carrying a bus card on Shabbat (with an eruv) to use after Shabbat — “No, you can’t do anything for after Shabbat, it is hachana.” Filling the fridge with water bottles for the week — “No, you can’t, it’s הכנה. Bedieved you can use it.”
Source: Carrying a Bus Card for After Shabbat · Putting Bottle in Fridge for After Shabbat