How can I sort a Map based upon on the size of its Collection values?

You can simply map the hash to list of Pair() and then sort the list

val map: MutableMap<String, List<String>> = HashMap()

map["USA"] = listOf("CA", "IA", "IL")
map["India"] = listOf("MUM", "CAL")
map["Canada"] = listOf("TOR")

val sortedPairs = map.map { Pair(it.key, it.value) }.sortedBy { it.second.size }.toMap()

println("sorted List: $sortedPairs")

HashMap does not have a guaranteed iteration order so you will need to collect to a LinkedHashMap in order for the sorting to be meaningful.

import static java.util.Comparator.comparingInt;
import static java.util.stream.Collectors.toMap;

Map<String, List<String>> sorted = map.entrySet().stream()
    .sorted(comparingInt(e -> e.getValue().size()))
    .collect(toMap(
        Map.Entry::getKey,
        Map.Entry::getValue,
        (a, b) -> { throw new AssertionError(); },
        LinkedHashMap::new
    )); 

The AssertionError is thrown because a combiner function is only used on parallel streams, which we are not using.

You can also use comparingByValue if you find it more readable:

import static java.util.Map.Entry.comparingByValue;

Map<String, List<String>> sorted = map.entrySet().stream()
    .sorted(comparingByValue(comparingInt(List::size)))
    // ... as above

You have two problems.

1. Map doesn't support sorting.

2. SortedMap doesn't support sorting on values only sorting on keys.

As a result of this using a Map or SortedMap isn't going to help you. What you need to do is iterate over you map and put each Entry<String, ArrayList<String>> into a collection such as a List and then sort the list with a custom compare. See this example TreeMap sort by value or this example Sorting LinkedHashMap