Either side can play for a long ko at A1/B1 (a throw-in at A1 for Black, B1 for White), but often it’s disadvantageous for one side to fill one the remaining liberties in such a situation (at C3 or D1), as the other side always takes the ko first and puts one’s stones into atari after that (so the burden is on the player who fills the internal extra liberties to find a first ko threat, and it makes it easier for the other side to resolve the seki/ko).
Black loses points by filling these liberties if White wins the ko, and White puts the entire group in danger of capture by filling one of the liberties.
But depending on the ko threat situation, it can be very playable for one side or the other to throw in at A1 or B1.
To avoid the ko, if it benefits Black too much, White can play A1.
Similarly, Black can play B1 if White throwing in at B1 is a problem.
(which settles the situation into a seki for certain)
Edit: Black can also fill the ko after beginning it, instead of filling the remaining internal liberty, which settles the situation into a seki as well.)
There are “status problems” for this, which are problems in which a group may be unsettled (killable or liveable) or already settled (either alive or dead regardless of who plays next). ^^
There is a practice game with relatively simple status problems on this server :
Another technique is to simply find board positions or corners, from one’s games or others’ (such as professional games), and to read a corner or side position as if it were a status problem (what happens when White plays first, and what happens if Black plays first). 
There are also tsumego of the sort in books like Trouble Master by In Seong Hwang 8d, which emphasise building an ability for finding “troubles” (aji) in unsettled areas in a board position, without necessarily being given a clear goal such as ‘Black to kill’ or being told where these unsettled areas/aji are.
(In a real game, there are no labels of ‘White to live’ or ‘Black to find a tesuji’, so finding the aji and possibilities at any point of the game is one part of a player’s strength.)
(It’s an excellent book of what tsumego I have solved in it, and I really enjoy the concept, for anyone interested.)
For tsumego, one also can try reading the status of an interesting group in one’s games at each point in the development. (for example before the killing/living attempt, then after move 1, then after move 2, after move 3, 4, etc. of the sequence in the game)
Or one can practise looking for aji similarly, by simply reading positions in real games, looking at the position before one missed something, and also searching for aji which wasn’t played out at any particular point.