I am looking for datasets that are used as a testing standard in the fully connected neural networks (FCNN). For example, in the image recognition and CNN, CIFAR datasets are used in most of the papers, but can't find anything like that for the FCNN.
You can use MNIST obviously but I'd also suggest you have a look at UC Irvine's datasets: https://archive.ics.uci.edu/ml/datasets.php
This answer doesn't provide a declaration as to which dataset(s) are used quasi-ubiquitously in research/literature. It simply provides a frame-of-reference for where to look for structured datasets and examples of two structured datasets that could be used in general.
You want to look for structured datasets.
Good examples of this are things like housing price datasets.
As an answer directly though, try a housing price dataset like the Boston Housing Price Dataset.
You could also use the famous Titanic Dataset on Kaggle.