All speeds will be reduced to 0.
Specifically you cannot fly when your speed is 0:
PHB p. 191
If a flying creature is knocked prone, has its speed reduced to 0,
or is otherwise deprived of the ability to move, the creature falls,
unless it has the ability to hover or it is being held aloft by magic,
such as by the fly spell.
As for other miscellaneous types of movement, they also depend on speed:
PHB p. 182
Special Types of Movement
Movement through dangerous dungeons or wilderness areas often involves more than simply walking. Adventurers might have to climb, crawl, swim, or jump to get where they need to go.
Climbing, Swimming, and Crawling
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. At the GM’s option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check. Similarly, gaining any distance in rough water might require a successful Strength (Athletics) check.
Since these depend on movement, if you can't move because your speed is 0 you cannot preform them.
In general, all of your speeds are connected and when you use part of your speed for one type of movement, it subtracts it from all. The PHB has a good example of this:
PHB p. 190
Using Different Speeds
If you have more than one speed, such as your
walking speed and a flying speed, you can switch back and forth
between your speeds during your move. Whenever you switch, subtract
the distance you've already moved from the new speed. The result
determines how much farther you can move. If the result is 0 or less,
you can’t use the new speed during the current move.
For example, if
you have a speed of 30 and a flying speed of 60 because a wizard cast
the fly spell on you, you could fly 20 feet, then walk 10 feet, and
then leap into the air to fly 30 feet more.