Generally, above ground pools do not affect home values for appraisal or tax assessment purposes. Note that inground pools generally do increase home values, typically in the 5-8% range. That being said...
Scenario 1: Suppose when you purchased your home, that you negotiated with the seller that you would pay $500 more for the above ground pool. Further suppose you sold said pool for $800. Technically, your home value's basis should have been $500 lower, and you would have $300 in income in the year you sold the pool.
Scenario 2: Suppose when you negotiated the sale of the house, the seller said you can have the above ground pool for free, if you want it. (In other words, if you said you didn't want it, the seller would have removed it before you closed on the house without changing the agreed upon price.) An initial assumption might be that since you paid $0, and sold it for, say $800, that the full amount would now be taxable. However, instead, this would have (arguably) been considered a gift to you at the pool's market value at the time you purchased the home. Assuming the value did not increase by much, or at all since then, you would not need to treat the proceeds from the sale as income.
Clarification: In scenario #2, I claim the pool is "arguably" a gift because it's difficult to mix gifts together with other sales or trades. If you were audited over it, you might need to prove that the price of the home was going to be the same regardless of whether you elected to keep the pool. Otherwise the price you paid for the home would technically have been lower than the sale price. One way to approach these sort of classifications is to imagine that the auditor is omniscient, and then decide if they would agree with your classification.
