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I'm looking for sources of information freely available online about happenings in London the day July 1st 1858 (besides the famous Darwin-Wallace presentation, that is.)
I have found copies of the front page of The Times, but they are almost illegible and mostly full of advertising.

The first place that I would search is the British Newspaper Archive. Note that this site requires a subscription to actually view the newspapers, although it is free to search.

If you have a subscription, it is possible to zoom in on the high-resolution scans of the newspapers, making it easier to read the stories than it is with many of the originals!

Many libraries in the UK offer free access to the British Newspaper Archive online. Access is also available with some subscriptions to the FindMyPast genealogy site.

Use the Advanced search function and set the publication place as 'London' and choose an appropriate date-range.

Bear in mind that events occurring on 1 July may not appear in the newspapers until several days (or even weeks)) later. You should also remember that not every paper was published daily. Your date-range should reflect that.

It may take several iterations to refine your search until you are satisfied with the results.

For example, a simple search limited to the exact date 1 July 1858 gives 11 hits.

Extending that to include 1 & 2 July 1858 gives 23 hits.

The titles include:

• London Daily News


• London Evening Standard


• Morning Advertiser


• Morning Chronicle


• Morning Post


• Evening Mail


• West Middlesex Advertiser and Family Journal


• The Globe


• Police Gazette


• County Courts Journal


• Public Ledger and Daily Advertiser


• Shipping and Mercantile Gazette


• Lloyd's List

If, for whatever reason you are unable to view these titles on the BNA, you at least now have some titles to search for on other sites. (Note that some of those sites may offer limited free access to new subscribers!). Wikipedia includes a (probably incomplete) List of online newspaper archives

Once you have a list of titles, it is always worth checking Google Newspapers to see if any appear in their digitised collections.

Also, you should be aware that it was common for the front page of newspapers to be dedicated to adverts with news inside.

A good resource is this page on the National Archive's site. It's a very useful page, listing several different ways you can read historic newspapers, such as websites and physical locales.

You may not be able to find much (many of the websites are paywalled or are for a different location), but there is some stuff. For example, here's some news about bankruptcy from The London Gazette:

WHEREAS a Petition for adjudication of Bankruptcy, was, on the 1st day of July, 1858, filed in Her Majesty's Court of Bankruptcy in London, against Joseph Stratford, of No. 20, Pelham-street, Thurlow-square, Brompton, in the county of Middlesex, Baker, and he having been declared bankrupt, is hereby required to surrender himself to Robert George Cecil Fane, Esq., one of Her Majesty's Commissioners of the Court of Bankruptcy, on the 15th of July instant, at eleven in the forenoon precisely, and on the 13th day of August next, at twelve o'clock at noon precisely, at the Court of Bankruptcy...

I'm looking for sources of information freely available online about happenings in London the day July 1st 1858

Consider going to your local library or university, and enquiring about where you can find London news archives (be it online or offline) from that day.