Regular expression for any binary string:

REGEXP: [01]*
ACCEPT: 0
ACCEPT: 1
ACCEPT: 00
ACCEPT: 10
ACCEPT: 111010101010111
REJECT: cat
REJECT: 123
REJECT: he110, w0r1d

Regular expression for any binary string that represents an unsigned integer that is EVEN:

REGEXP: [01]*0
ACCEPT: 0
ACCEPT: 10
ACCEPT: 1110
ACCEPT: 010
ACCEPT: 00011000110110
REJECT: 1
REJECT: 011
REJECT: 0101

Regular expression for any binary string of EVEN length:

REGEXP: ([01]{2})*
ACCEPT: 00
ACCEPT: 01
ACCEPT: 1010
ACCEPT: 0001
ACCEPT: 0001101100011011
REJECT: 0
REJECT: 1
REJECT: 111
REJECT: 010
REJECT: 0001101

Regular expression for any binary string that contains the pattern 0110 OR the pattern 1001:

REGEXP: .*(0110|1001).*
ACCEPT: 0110
ACCEPT: 1001
ACCEPT: 01101001
ACCEPT: 011001
ACCEPT: 111101101111
ACCEPT: 0000010010000
REJECT: 0
REJECT: 1
REJECT: 0101
REJECT: 1010
REJECT: 1100010001110

Regular expression for any binary string that contains the pattern 0110 AND the pattern 1001:

REGEXP: .*((1001.*0110)|(0110.*1001)|011001|100110).*
ACCEPT: 01101001
ACCEPT: 000110001001000
ACCEPT: 011001
ACCEPT: 100110
ACCEPT: 000011001000
REJECT: 0
REJECT: 1
REJECT: 0110
REJECT: 1001
REJECT: 01101
REJECT: 10010
REJECT: 1100010001110

Regular expression that will accept three or four words:

REGEXP: ([\w]*\s){2,3}[\w?]*
ACCEPT: one two three
ACCEPT: one two three four
ACCEPT: 1 2 3 4
ACCEPT: how many words?
REJECT: word
REJECT: word word
REJECT: word word word word word

Regular expression that will accept the word cat and the word hat with at most 2 other words between them:

REGEXP: cat\s(\w*\s){0,2}hat
ACCEPT: cat hat
ACCEPT: cat in hat
ACCEPT: cat in the hat
REJECT: hat cat
REJECT: cathat
REJECT: catinthehat
REJECT: cat in the phat hat
REJECT: cat in the super phat hat

Regular expression that will accept a properly formatted 24-hour time (0:00 ... 23:59):

REGEXP: ((1?[0-9])|(2[0-3])):[0-5][0-9]
ACCEPT: 12:34
ACCEPT: 0:00
ACCEPT: 23:59
ACCEPT: 9:00
ACCEPT: 10:00
ACCEPT: 11:11
ACCEPT: 15:15
REJECT: 24:00
REJECT: 12:32 pm
REJECT: 0:60
REJECT: 9:99
REJECT: 04:00
REJECT: 4
REJECT: -4:00

Regular expression for a binary string that has no consecutive 0's and no consecutive 1's:

REGEXP: ((01)*0?)|((10)*1?)
ACCEPT: 1
ACCEPT: 0
ACCEPT: 10
ACCEPT: 01
ACCEPT: 101
ACCEPT: 010
ACCEPT: 1010
ACCEPT: 0101
ACCEPT: 10101
ACCEPT: 01010
ACCEPT: 101010
ACCEPT: 010101
REJECT: 00
REJECT: 11
REJECT: 001
REJECT: 100
REJECT: 110
REJECT: 011
REJECT: 1001
REJECT: 0110
REJECT: 01010100101010
REJECT: 10101011010101

Regular expression for a binary string that represents an unsigned value that is divisible by 3:

REGEXP: (1(01*0)*1|0)*
ACCEPT: 11
ACCEPT: 110
ACCEPT: 1001
ACCEPT: 1100
ACCEPT: 10101
ACCEPT: 11011
ACCEPT: 100001
ACCEPT: 111111
ACCEPT: 1110101
ACCEPT: 101010110
ACCEPT: 000111000101
REJECT: 1
REJECT: 10
REJECT: 100
REJECT: 111
REJECT: 110001
REJECT: 1000100
REJECT: 10010010
REJECT: 10101100
REJECT: 11100101
REJECT: 11011101
REJECT: 110110111