Return to Answer

You're looking for the First Monoid instance for Maybe. Used like—

import Data.Foldable (foldMap)
import Data.Monoid (First(..))
fractionParse :: String -> FractionParse
fractionParse s = fromMaybe Rubbish . getFirst
 $ foldMap First [ maybeFraction s
 , maybeWhole s
 ]

It doesn't make sense to encode your own failure values in your datatypes, so I'd remove the Rubbish Constructor.

data FractionParse = Fraction Int Int | Whole Int
 deriving (Show, Eq)
fractionParse :: String -> Maybe FractionParse

Use Control.Monad.guard instead of manually sending up your own sentinel values (as in maybeNumDenom).

maybeNumDenom s = do
 guard $ containsExactlyOneSlash s
 -- ...

Take advantage of incremental parsing to implement maybeNumDenom, then you won't have to do so much extraneous filtering, counting, and finger crossing. And by removing the Rubbish constructor, at the point you have a correct parse you know you can return a Fraction, so—

import Data.Maybe (listToMaybe)
maybeFraction :: String -> Maybe FractionParse
maybeFraction s = listToMaybe $ do
 (n, '/':s') <- reads s
 (d, '') <- reads s'
 return (Fraction n d)

The above function operates in the list monad. listToMaybe converts a list to a Maybe value by returning Just the first element of the list, or Nothing in the case of an empty list. reads :: Read a => String -> [(a, String)] produces possible parses from a given string, returning the remainder of the unparsed string as the second element of each tuple. Binding to (d, '') ensures that only parses that consume the whole string will be returned.

You're looking for the First Monoid instance for Maybe. Used like—

import Data.Foldable (foldMap)
import Data.Monoid (First(..))
fractionParse :: String -> FractionParse
fractionParse s = fromMaybe Rubbish . getFirst
 $ foldMap First [ maybeFraction s
 , maybeWhole s
 ]

It doesn't make sense to encode your own failure values in your datatypes, so I'd remove the Rubbish Constructor.

data FractionParse = Fraction Int Int | Whole Int
 deriving (Show, Eq)
fractionParse :: String -> Maybe FractionParse

Use Control.Monad.guard instead of manually sending up your own sentinel values (as in maybeNumDenom).

maybeNumDenom s = do
 guard $ containsExactlyOneSlash s
 -- ...

Take advantage of incremental parsing to implement maybeNumDenom, then you won't have to do so much extraneous filtering, counting, and finger crossing. And by removing the Rubbish constructor, at the point you have a correct parse you know you can return a Fraction, so—

import Data.Maybe (listToMaybe)
maybeFraction :: String -> Maybe FractionParse
maybeFraction s = listToMaybe $ do
 (n, '/':s') <- reads s
 (d, "") <- reads s'
 return (Fraction n d)

The above function operates in the list monad. listToMaybe converts a list to a Maybe value by returning Just the first element of the list, or Nothing in the case of an empty list. reads :: Read a => String -> [(a, String)] produces possible parses from a given string, returning the remainder of the unparsed string as the second element of each tuple. Binding to (d, "") ensures that only parses that consume the whole string will be returned.