There are multiple “finally” in Japanese, each with different connotations and usage.
やっと
やっと usually implies the result is belated, and it should’ve happened earlier and/or more easily.
- 隣のうるさい犬がやっと吠え止んだ -> The annoying next-door dog finally stopped barking
- 家の前の道路工事がやっと終わった -> The road constructions in front of my house finally ended
- 彼はやっと運転免許試験に合格した -> He finally passed the driving-license exam
However it also emphasizes a sense of relief.
- 雨がやっと止んだ -> The rain finally stopped
- やっと食べ物にありついた -> I finally found something to eat
漸く（ようやく）、ようやっと
漸く and ようやっと (which was made from ようやく + やっと) works basically the same with やっと. 漸く is more literary than やっと, and ようやっと is more emphatic than やっと
- 彼は五回目でようやっと運転免許試験に合格した -> He finally passed the driving-license exam in his fifth try
遂に（ついに）
遂に implies something has finally happened after a long time and a lot of efforts, difficulties, unexpected incidents, etc..
Usually 遂に is used when the result is triumphant, hard-won.
- 彼は遂に夢を叶えた -> He finally realized his dream
However it can be used that when the result is negative but very dramatic.
- 彼は大勢の敵と勇敢に戦ったが、遂に力尽きた -> He fought bravely while outnumbered, but finally got exhausted (and died)
遂に is not used when annoyance which you can’t do anything about ends on its own.
隣のうるさい犬が遂に吠え止んだ家の前の道路工事が遂に終わった
However you can use 遂に if you worked hard to end the annoyance.
- 彼のことを拒否し続けていたら彼は遂に諦めた -> I kept rejecting him and he finally gave up
いよいよ
いよいよ is some event that has been long-awaited is finally about to happen.
- ロケットの発射はいよいよ明日だ -> The rocket launch is finally tomorrow
いよいよ is not about the end result. Rather, and it is about an event that decides the end result is going to happen soon. So the following is wrong.
彼はいよいよオリンピックで金メダルを取った
Also いよいよ is never used for something to stop, no matter how long-awaited it is.
隣のうるさい犬がいよいよ吠え止んだ
到頭（とうとう）
到頭 is used when the end result is usually negative, yet somewhat expected.
- オイルを交換せずに使っていたらエンジンが到頭壊れた -> I kept driving without changing the oil and the engine finally broke
到頭 can be used when the end result is negative, despite a lot of efforts.
- 犯人は到頭見つからなかった -> In the end the perpetrator was never caught
仕舞いに（しまいに）
仕舞いに is similar to 到頭. You persist in doing something, and the end result is usually negative but foreseeable.
- オイルを交換せずに使っていたら仕舞いにエンジンが壊れた -> I kept driving without changing the oil and the engine finally broke
仕舞いに is different from 到頭 in that it can NOT be used as “the end result is negative despite a lot of efforts”.
犯人は仕舞いには見つからなかった