As a friend who recently arrived in China said, "Yes but do they really USE all of those measure words?" I don't know about all. But yes, unfortunately, they do use a great deal of them.
These are all the measure words I have heard in natural conversation. Maybe not worth memorizing, but as one learns to speak Chinese, and especially for listening, they're certainly worth being able to recognize (I can get stumped pretty easily when I hear a seemingly out of place word). ll also give the context I heard the word used in.
If you want to know what other things that measure word applies to, I suggest consulting the back of Chubby or some other measure word reference.
Measure word > for what kind of thing
bǎ 把 > stuff with handles (knives), chairs
qǐng lái sān bǎ sháozi 请来三把勺子= please bring 3 spoons
bān 班 > scheduled transportation
zhè bānxiào chē shì bú shì zuì hòu yì bān? 这班校车是不是最后一班？= Is this the last school bus?
běn 本 > bound stuff (books, magazines)
wǒ yǒu sān běn cídiǎn 我有三本辞典= I have 3 dictionaries
bǐ 笔 > sums of money
wǒ bìxū huā yì bǐ qián 我必须花一笔钱= I have to pay a sum of money
bù 部 > cell phones, sequels of a book/movie
nǐ wèishénme yǒu liǎng bù shǒujī? 你为什么有两部手机？= Why do you have 2 cell phones?
duàn 段 > sections of long stuff, parts of songs
yí duàn fěnbǐ 一段粉笔= a piece of chalk
duǒ 朵 > flowers
I can't remember the sentence, but it was definitely about flowers.
fèn 份 > bundles/batches (servings of food, multi-page documents)
yào jǐ fèn? 要几份= How many servings (do you) want?
jiā 家 > stores
zhège chéngshì yǒu jǐ jiā wòěrmǎ? 这个城市有几家沃尔玛？= How many Walmarts does this city have?
jiān 间 > rooms, houses (one room)
zhège xuéxiào de sùshè jǐ gè rén yì jiān? 这个学校的宿舍几个人一间 = How many people are there in one dorm room at this school?
jiàn 件 > shirts
yí cì zuì duō zhǐ néng ná sān jiàn 一次最多只能拿三件= the most you can take at one time is 3 (shirts)
*what the girl at the fitting rooms said
jiǎng 讲 > long (usually 2 hour) class periods that may include a break (as opposed to "jié")
jīntiān nǐ yǒu jǐ jiǎng kè 今天你有几讲课？= How many classes do you have today?
jié 节 > 1 hour class periods
jīntiān nǐ yǒu jǐ jié kè？今天你有几节课？= How many hours of class do you have today?
jù 句 > sentences
wǒ zhǐ huì shuō yí jù Kūnmíng huà 我只会说一句昆明话= I can only say one sentence of the Kunming dialect
kē 棵 > trees
jīntiān wǒmen zhòng le èrshí kē shū 今天我们种了二十棵树= today we planted 20 trees
kǒu 口 > people, family members
nǐ de jiā yǒu jǐ kǒu rén 你的家有几口人？= How many people are there in your family?
*you could also use "ge 个"
kuài 块 > chunks of tofu/yuan money
ná yí kuài dòufu 拿一块豆腐= I'll take one piece of tofu
lèi 类 > kinds of people
wǒmen shì yí lèi rén 我们是一类人= we are (the same) one kind of person
lì 粒 > small round stuff
wǒ shènzhì bù néng chī yí lì huāshēng 我甚至不能吃一粒花生= I can't even eat one peanut
liàng 辆 > vehicles with wheels (but not trains)
wǒ yǒu liǎng liàng zìxíngchē 我有两辆自行车= I have 2 bicycles
mén 门 > subjects (academic)
nàge kǎoshì yǒu sān mén 那个考试有三门= that test has 3 subjects
*e.g. math, English, history
miàn 面 > mirrors
wǒ xiǎng mǎi yí miàn xiǎo jìngzi 我想买一面小镜子= I want to by a small mirror
pái 排 > rows of seats, lines of people
zhàn chéng yì pái! 站成一排！= line up! [stand become 1 line]
pǐ 匹 > horses
yì pǐ mǎ gòulái le 一匹马过来了= a horse came over
*from a children's story
piàn 片 > slices, expanses (like clouds, beaches)
yí piàn ròu 一片肉= a slice of meat
shēng 声 > utterances, sounds
wǒ yào xiān duì nǐ shuō yì shēng "xièxie" 我要先对你说一声"谢谢"= I want to say "thanks" in advance
shǒu 首 > songs, poems
zhè shǒu gē shì shéi de? 这首歌是谁的？= Who's song is this?
*this is how I ask about the music blasting over the speakers on campus
shuāng 双 > pairs (chopsticks, shoes, eyes)
wǒ yào mǎi sān shuāng tuōxié 我要买三双拖鞋= I want to buy 3 pairs of slippers.
tái 台 > big electric stuff, pianos
wǒ yǒu yì tái xǐyījī 我有一台洗衣机= I have a washing machine
tào 套 > apartments, houses (multiple rooms)
nàbian de fángzi yào duōshao qián yí tào? 那边的房子要多少钱一套？= Those houses cost how much per house?
tiáo 条 > roads, long stuff (rivers, snakes, pants)
zhè tiáo lù jiào shénme míngzi 这条路叫什么名字？= What's this road called?
tóu 头 > pigs, cows
yì tóu máoniú 一头牦牛= one yak
wèi 位 > people (polite)
jǐ wèi 几位= How many people?
*in your party at a restaurant
zhāng 张 > flat stuff (pieces of paper, tables, CDs)
ná shí zhāng kòngbái de CD 拿十张空白的CD= I'll take 10 blank CDs
zhī 只 > cats, birds
"liǎng zhī lǎohǔ 两只老虎" = "2 tigers" (sung to the tune of "Frere Jacques")
zhī 只 > one of a pair
bā zhī yǎnjing 八只眼睛= 8 Eyes
*the name of a 4 person singing group
zhī 支 > stick-like stuff (pen, chopstick, shoe, eye)
yì zhī bǐ bú gòu 一支筷子不够= 1 pen isn't enough
zhǒng 种 > types of stuff
yǒu liǎng zhǒng júzi 有两种桔子= there are 2 kinds of oranges
zuò 座 > mountains
zhōngguó zuì yǒumíng de sān zuò shān 中国最有名的三座山= China's 3 most famous mountains