In North America all major brokerage firms provide their clients with on-line services, they ALL provide access to research, trading platforms and operational services. .
There is no "Most" trustworthy firm, Customer need to select the site that is best for them. Traders have requirements in a site, while investors have other requirements.
Although most sites are geared to general securities options and commodities, however customers may have special needs for the types of products they trade, and the markets in which they trade. For example, I use Fidelity for investing. TDAmeritrade for equities and & options trading while mutual funds are done through Vanguard.
It seems that the most popular firms for on-line investing/trading all of which provide excellent platforms and services are; Scottrade; Chas.Schwab; TDAmeritrade; Fidelity; E-Trade, Edward Jones, Vanguard, and T.Rowe Price.
50+ years of experience