# [Ohm], 35 bytes

    l┼xl2+┼DRîsRl≥al≤Dla°┼îa/ì\?≥;+WD╤k

[Try it online!][TIO-j3g2zrcx]

Outputs 0 for decimal, 1 for hex and 2 for scientific.

Explanation:

    l                                      Implicit input, get length                                          
     ┼                                     Input again
      x                                    To hex
       l                                   Get length
        2+                                 Add 2 because of "0x"
          ┼                                Get input again
           D                               Duplicate on the stack
            RîsR                           Remove zeroes at the end (reverse, to int, to string, reverse)
                l                          Get length (= length of base)
                 ≥                         Add 1 because to count "e" in the scientific notation
                  a                        Swap top two values on the stack
                   l≤                      Get length - 1 ( = get the exponent of 10 in scientific notation)
                     D                     Duplicate on the stack
                      l                    Get length ( = length of the exponent)
                       a                   Swap. Now on top of the stack we have the exponent again
                        °                  10^exponent
                         Ō                Get input for the fourth time
                           a/              Divide input by the 10^exp calculated earlier
                             ì\?           If this thing is not an integer...
                                ≥;         ...add one to count the "."
                                  +        Sum base length ( + "e") + exponent length ( + ".")
                                   W       Wrap stack in array
                                    D      Duplicate
                                     ╤k    Get index of min value
                           

[Ohm]: https://github.com/MiningPotatoes/Ohm
[TIO-j3g2zrcx]: https://tio.run/##y8/I/f8/59GUPRU5RtpAyiXo8LrioJxHnUsTgcQSl5zEQxuAwofXJeofXhNjDxS31g53eTR1Sfb//4YGCGAIAA "Ohm – Try It Online"