There is no completely clear answer to this question because there is no one exact methodology to measure and rate the partisan bias of news sources.
Here are a couple of resources that can help:
The Allsides Bias Ratings page allows you to filter a list of news sources by bias (left, center, right).
AllSides uses a patented bias rating system to classify news sources as left, center, or right leaning. Components of the rating system include crowd-sourcing, surveys, internal research, and use of third party sources such as Wikipedia and research conducted by Groseclose and Milyo at UCLA. Note that while the Groseclose & Milyo results are popular, the methodology it is not without critique.
Note that AllSides does periodically revisit classifications for particular sources and their ratings may change over time. In some instances, the News and Opinion sections of a specific news outlet may be rated separately (for examples, see Wall Street Journal, or NPR).
A report based on a 2014 survey shows which news sources are used and considered trustworthy based on individual's political values (liberal or conservative). Note that this report measures the political leanings of the audience rather than the source itself.
The Blue Feed, Red Feed tool relies on data from a large study of Facebook users conducted in 2015 by Bakshy, Messing, & Adamic (U-M Library access). The ideological alignment of news sources on Facebook was measured based on the stated ideological affiliations of users who interacted with the content.
Here are a few examples of major news sources and their so-called "bias" based on ratings from AllSides (as of March 2017) and the reported level of trust from partisan audiences from the Pew Research Center survey.
Note that much of these ratings are based on surveys of personal perceptions. Consider that these may be impacted by the hostile media effect, wherein "partisans perceive media coverage as unfairly biased against their side" (source: Perloff, R. M. (2015). A three-decade retrospective on the hostile media effect. Mass Communication and Society, 18(6), 701-729. (U-M Library access)).
The Wall Street Journal news content has a bias rating of "Center" according to AllSides (although the WSJ Opinion section is rated as "Lean Right" - the news and opinion content are rated separately). The Pew Research Center found that the Wall Street Journal is read by people of all political leanings and is the only news source that is more trusted than distrusted by people all across the political spectrum. There is, however, some debate but it has been accused of being biased to both the left and the right.
The BBC News has a bias rating of "Center" from AllSides. According to Pew, BBC is more trusted than distrusted among the different ideological groups except for consistently conservative individuals who view BBC as about equally trusted as distrusted.
The following news sources are rated as "Center" by AllSides and are noted for stating an independent and unbiased agenda, as well as consistently providing balanced coverage (giving time to both sides of an issue). However, they are listed here as almost center because of audience distrust from conservatives.
Respondents to the Pew survey who are consistently liberal, mostly liberal, and mixed views, rated these sources as more trusted than distrusted. Mostly conservatives rated them as about equally trusted as distrusted; Consistently conservatives rated them as more distrusted than trusted.
A key finding from the Pew survey is that conservatives have a higher level of distrust overall of news sources and consume a much narrower range of news sources. Liberal audiences trust and consume a broader range of news sources.
Therefore, there are more news sources that are rated or perceived as center-left/slight left leaning (especially mainstream sources); there are fewer news sources that are center-right. This represents the popular conservative discourse that the mainstream media has a liberal bias (and potential hostile media effect).
These news sources are rated as "Lean Left" by Allsides. According to the Pew survey, they are more distrusted than trusted by people who are mostly conservative and consistently conservative; and are more trusted than distrusted by people who are mixed, mostly liberal, and consistently liberal.
USA Today was moved from "Center" to "Lean Left" in July 2021 after an AllSides Editorial Review.
These news sources are rated as "Left" by Allsides. According to the Pew survey, they are more trusted than distrusted by people who are mostly liberal, consistently liberal; and are more distrusted than trusted by people who are mostly conservative, and consistently conservative.
These news sources are rated as "Right" by Allsides. According to the Pew survey, they are more distrusted than trusted by people who are mostly liberal and consistently liberal; and are more trusted than distrusted by people who are mixed, mostly conservative, and consistently conservative. Note that this represents a further bias leaning than the "towards the left" sources. There are no direct parallels on the right.
These news sources are rated as "Right" by Allsides. According to the Pew survey, they are more distrusted than trusted by people who are mostly liberal, consistently liberal; and are more trusted than distrusted by people who are mostly conservative, and consistently conservative.