Integer-Digits of the Arithmetic-Tables

JavaScript (ES7), 128 bytes

f=
c=>[...c+`0123456789`].map((r,_,a)=>a.map(l=>l==c?r:r==c?l:/^\d$/.test(l=c<`^`?eval(l+c+r):l|c?l**r:l/r)?l:` `).join` `).join`
`

<select onchange=o.textContent=f(this.value)><option>><option>+<option>-<option>*<option>/<option>%<option>^<option>&<option>,<option>.</select><pre id=o>

Special-casing 0^0 cost me 8 bytes.

Japt, 45 bytes

Collaborated with @ETHproductions

AÆAÇU¥'p«Z«XªOvZ+U+X)+P r"..+"SÃuXÃuAo uU)m¸·

Run it online!

Takes input as:

"+" for addition

"-" for subtraction

"*" for multiplication

"/" for division

"p" for exponentiation

"%" for modulo

Explanation (With expanded shortcuts):

AÆ  AÇ  U¥ 'p«  Z«  Xª OvZ+U+X)+P r"..+"SÃ uXÃ uAo uU)m¸  ·
AoX{AoZ{U=='p&&!Z&&!X||OvZ+U+X)+P r"..+"S} uX} uAo uU)mqS qR

A                                                             // By default, 10 is assigned to A
 o                                                            // Create a range from [0...9]
  X{                                         }                // Iterate through the range, X becomes the iterative item
    Ao                                                        //   Create another range [0...9]
      Z{                                 }                    //   Iterate through the range, Z becomes the iterative item
                                                              //     Take:
        U=='p                                                 //       U (input) =="p"
             &&!Z                                             //       && Z != 0
                 &&!X                                         //       && X != 0
                     ||                                       //     If any of these turned out false, instead take
                       Ov                                     //       Japt Eval:
                         Z+U+X                                //         Z{Input}X
                              )+P                             //     Whichever it was, convert to a string
                                  r"..+"S                     //     Replace all strings of length 2 or more with " "
                                                              //     (this makes sure the result !== "false" and has length 1)
                                           uX                 //   Insert X (the row number) into the front of the row
                                               u              // Insert at the beginning the first row:
                                                Ao            //   [0...9]
                                                   uU)        //   with the input inserted at the beginning
                                                      mqS     // Join each item in the final array with " "
                                                          qR  // Join the final array with "\n"

Operation Flashpoint scripting language, 343 333 303 301 bytes

f={o=_this;s=o+" 0 1 2 3 4 5 6 7 8 9\n";i=0;while{i<10}do{j=0;s=s+format["%1",i];if(i<1&&(o=="/"||o=="%"||o=="^"))then{if(o=="^")then{if(j<1)then{s=s+"  ";j=1}}else{s=s+"\n1";i=1}};while{j<10}do{r=call format["%1%2%3",j,o,i];if(r>9||r<0||r%1>0)then{r=" "};s=s+format[" %1",r];j=j+1};s=s+"\n";i=i+1};s}

Call with:

hint ("+" call f)

Ungolfed:

f=
{
    o=_this;
    s=o+" 0 1 2 3 4 5 6 7 8 9\n";
    i=0;
    while{i<10}do
    {
        j=0;
        s=s+format["%1",i];
        if(i<1&&(o=="/"||o=="%"||o=="^"))then
        {
            if(o=="^")then{if(j<1)then{s=s+"  ";j=1}}
            else{s=s+"\n1";i=1}
        };
        while{j<10}do
        {
            r=call format["%1%2%3",j,o,i];
            if(r>9||r<0||r%1>0)then{r=" "};
            s=s+format[" %1",r];
            j=j+1
        };
        s=s+"\n";
        i=i+1
    };
    s
}

Output:

operator +

operator -

operator *

operator /

operator ^

operator %