Much of my postgraduate research focused on the Irish Brigade in French service in the 18th century, so I apologise for the somewhat Hiberno-philic answer I am about to give. If any experts on Prussian, Swedish or Russian doctrine care to chime in I would love to hear what they have to say, as what is below is hardly comprehensive!

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First, a bit of background.

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The bayonet's development was combined with the phasing out of pike infantry (you can argue a correlation v causation here, but that is an answer for another time), as when a sharp object was added to a musket, you essentially have a rudimentary pike, making the weapon redundant.

That being said, the pike's purpose at the dawn of the 18th century was primarily defensive. It was protection against cavalry, as offensive pike tactics were gradually phased out due to the wider availability of gunpowder weapons and cannon. The history of the Swiss Pikemen and their Landschneckt rivals tracks this nicely, and come the 18th century, you rarely see pikemen on the battlefield at all. Most infantry regiments would have either bayonets or in some exceptional cases, swords (Scottish highlanders being a good example of this). The one notable exception was the army of Sweden, who retained pikemen into the 18th century.

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Now, on to the question at hand. You are confronted with an advancing wall of bayonets; what do you do? First of all, chances are you are fighting against the French. French military doctrine at this time focused on firing, and closing the distance with the enemy. While the Dutch (and subsequently the British via William of Orange) adopted a platoon fire system in order to maximize the output of their weapons at range, the French doggedly believed that firing by rank and advancing was more effective. This is illustrated in the injuries sustained by their soldiers. Often, the Invalides record of a soldier's wound will state exactly where it is, and Eoghan Ó hAnnracháin argues that the lack of wounds in the left legs and sides of soldiers in France is indicative of the fact that they rarely stood side on, as one would be angled in a prolonged firefight. In addition, the French would be consistently out-shot by equivalent British or Dutch regiments, so charging with a bayonet would be far more effective. This is reflected in the account of one Captain Parker of the British Royal Irish Regiment, who describes how his men obliterate a similar Irish regiment in French service at the Battle of Malplaquet. Given this mis-match, it makes sense for the French army to favour closing the distance.

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How common was this kind of engagement during the 18th century? Generally speaking they were not uncommon in certain theatres, and most of the major battles of the War of the Spanish Succession in particular would feature at least one close-combat engagement, often involving Irish soldiers (we had a certain expertise in the area it seems). As the years wore on, and Early Modern Warfare continued to develop, there are other examples of bayonet v bayonet actions taking place, the most noticeable of which was the battle of Fontenoy (1745). The climax of this engagement saw a thick column of British (and Hannoverians, but I am going to refer to them as British) infantrymen advancing towards French lines, blowing away successive ranks of French and Allied soldiers, until finally they are halted by a bayonet charge by the Irish Brigade, grapeshot, and a mass French cavalry attack.

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What makes Fontenoy so important for this discussion is that the Irish did not bother to shoot at the British, instead voluntarily closing the distance. A quote from Charles O'Brien, Viscount Clare "Do not shoot until your bayonet is in their belly". What is even more interesting is that the Redcoats held their ground. They did fire a volley as you said (Clare's friend, Viscount Dillon was killed in this volley. Clare himself was knocked off his horse, but he opted to wear a breastplate and as such survived), but what followed was a vicious close-combat engagement. It wasn't a short, sharp shock, and the British only retreated after repeated cavalry charges from the French forced them back. And it was an orderly retreat, not a mass panic. So Fontenoy is one of the best examples we have of the way Bayonet v Bayonet engagements tended to work in the 18th century.

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The Peninsular campaign in the early 19th century sees yet another example of a close combat brawl. The battle of Barrosa is famous for an incident involving an Irish ensign by the name of Keogh capturing a French Eagle and raising it high in the melee to keep his men's spirits high. This battle consisted of one British division defeating two French ones, and in such an encounter moral is a key factor. As the French knew they outnumbered the British, they would obviously be more willing to remain in the fight.

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Another example of this moral factor can be seen in the Crimean War, at the battle of Inkerman (1854), where fog hid the Russian's clear numerical superiority from the British and French infantry, and thus they were willing to stay in a bayonet fight far longer than seems logical.

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So, to conclude: Would one side run at the sight of a bayonet charge? It is very possible. But well disciplined, and confident troops would not. Bayonet fights were fairly commonplace, but it wasn't a cheat-code in the manner your question implies. I hope this answered your question, and if you have any follow ups, feel free to ask away!

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Sources:

Parker, Robert Memoirs of the most remarkable military transactions between 1683-1718 with an account of several battles and sieges in Ireland and Flanders (Dublin, 1746) p. 138 accessed at http://www.askaboutireland.ie/aai-files/assets/ebooks/311%20Memoirs/311%20Memoirs%20of%20the%20Most%20Remarkable.pdf

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O’Callaghan, John Cornelius, The History of the Irish Brigades in the Service of France Glasgow, 1870.

O’Conor, Matthew, The Irish Brigades or Memoirs of the Most Eminent Irish Commanders who Distinguished themselves in the Elizabethan and Williamite Wars in their own country and in the service of France, Spain, etc. Dublin, 1885

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Anderson, M.S. The War of the Austrian Succession 1740-1748 London, 1995.

Canon. Power, P. "Bicentenary of Fontenoy: 1745-1945." Studies: An Irish Quarterly Review 34, no. 134 (1945): pp. 175-182.

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David, Saul., All The King's Men: The British Redcoat in the Era of Sword and Musket 2013

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Figes, Orlando The Crimean War: A History (New York, 2010)

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Holmes, Richard. Redcoat. The British Soldier in the Age of Horse and Musket London, 2002.

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Lynn, John A. Giant of the Grand Siècle: The French Army, 1610-1715 Cambridge, 1997.

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Murtagh, Harman. ‘Irish Soldiers Abroad 1600-1800’ in A Military History of Ireland Thomas Bartlett & Keith Jeffery (eds) Cambridge, 1996.

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Ó’hAnnracháin, Eoghan. ‘Casualties in the ranks of the Clare regiment at Fontenoy’ in Journal of the Cork Historical and Archaeological Society Vol. 99 (1994) pp. 96-110.

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Ó’hAnnracháin, Eoghan. ‘Guests of France: a description of the Invalides with an account of the Irish in that institution’ in Franco Irish Military Connections 1590-1945 ed. Nathalie Genet-Rouffiac & David Murphy Dublin, 2009.

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Ó’hAnnracháin, Eoghan. ‘Irish veterans at the Hôtel Royal des Invalides 1692-1769’ in The Irish Sword (1999) Vol. 21 p. 5-42.

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