The COVID-19 pandemic has affected millions of people globally, with respiratory organs being strongly affected in individuals with comorbidities. Medical imaging-based diagnosis and prognosis have become increasingly popular in clinical settings to detect COVID-19 lung infections. Among various medical imaging modalities, ultrasound stands out as low-cost, mobile, and radiation-safe imaging technology. In this comprehensive review, we focus on ultrasound-based AI studies for COVID-19 detection that use public or private lung ultrasound datasets. We surveyed articles that used publicly available lung ultrasound datasets for COVID-19 and reviewed publicly available datasets and organize ultrasound-based AI studies per dataset. We analyzed and tabulated studies in several dimensions, such as data preprocessing, AI models, cross-validation, and evaluation criteria. In total, we reviewed 42 articles, where 28 articles used public datasets, and the rest used private data. Our findings suggest that ultrasound-based AI studies for the detection of COVID-19 have great potential for clinical use, especially for children and pregnant women. Our review also provides a useful summary for future researchers and clinicians who may be interested in the field.